Proof. By [11] (see equation (14.6)), we may find constants \(a, b, C >0\) such that for any \(x \in \mathbb{P}(\mathbb{V})\) and any \(n \geqslant 1\), \[\begin{align} \mathbb{P} \left( d (g_n \cdots g_1 x, x_{g_n \cdots g_1}^M) > e^{-an} \right) \leqslant C e^{-bn}, \end{align}\] where, for \(g \in \mathbb{G}\), \(x_g^M \in \mathbb{P}(\mathbb{V})\) is the density point of \(g\), see page 224 of [11]. By the triangle inequality, we get that, for any \(x, x' \in \mathbb{P}(\mathbb{V})\), \[\begin{align} \mathbb{P} \Big( d (g_n \cdots g_1 x, g_n \cdots g_1 x') > e^{-an} \Big) \leqslant 2C e^{-bn}. \end{align}\] Using Fubini’s theorem and integrating over \(x'\) yield \[\begin{align} & \mathbb{E} \Big( \varrho \Big\{ x' \in \mathbb{P}(\mathbb{V}): d(g_n \cdots g_1 x, g_n \cdots g_1 x') > e^{ - an } \Big\} \Big) \notag\\ & = \int_{\mathbb{P}(\mathbb{V})} \mathbb{P} \Big( d (g_n \cdots g_1 x, g_n \cdots g_1 x') > e^{-an} \Big) \varrho(dx') \notag\\ & \leqslant 2C e^{-bn}. \end{align}\] By Chebyshev’s inequality, we obtain \[\begin{align} \mathbb{P} \left( \varrho \Big\{ x' \in \mathbb{P}(\mathbb{V}): d(g_n \cdots g_1 x, g_n \cdots g_1 x') > e^{ - an } \Big\} > e^{ - \frac{b}{2}n } \right) \leqslant 2C e^{ - \frac{b}{2}n }, \end{align}\] completing the proof of the lemma. ◻

Proof. As the function \(\varphi\) is Hölder continuous, there exist constants \(\alpha, C >0\) such that, for any \(x,x' \in \mathbb{P}(\mathbb{V})\), \[\begin{align} | \varphi(x) - \varphi(x') | \leqslant C d(x,x')^{\alpha}. \end{align}\] Taking the conditional expectation in 23 gives, for \(\mathbb{P}\)-almost all \(\omega=(g_1,g_2,\ldots) \in \Omega\), \[\begin{align} \mathbb{E} (\theta | \mathscr{A}_p)(\omega) = \int_{\mathbb{P}(\mathbb{V})} \varphi(g_1 \cdots g_p x) \nu(dx). \end{align}\] Letting \(a, b, c >0\) be as in Lemma 1, we obtain that, for any \(p \geqslant 1\), \[\begin{align} & \mathbb{E} \left| \theta- \mathbb{E} (\theta | \mathscr{A}_p) \right| \notag\\ &=\int_{\mathbb{P}(\mathbb{V})} \mathbb{E} \left| \varphi (g_p \cdots g_1 x)- \int_{\mathbb{P}(\mathbb{V})} \varphi (g_p \cdots g_1 x') \nu(dx') \right| \nu(dx) \\ & = \int_{\mathbb{P}(\mathbb{V})} \mathbb{E} \left| \int_{\mathbb{P}(\mathbb{V})} \mathbb{1}_{ \{ d(g_p \cdots g_1 x, g_p \cdots g_1 x') \leqslant e^{-a p} \} } \left( \varphi (g_p \cdots g_1 x) - \varphi (g_p \cdots g_1 x')\right) \nu(dx') \right| \nu(dx) \\ & \quad + \int_{\mathbb{P}(\mathbb{V})} \mathbb{E} \left| \int_{\mathbb{P}(\mathbb{V})} \mathbb{1}_{ \{ d(g_p \cdots g_1 x, g_p \cdots g_1 x') > e^{-a p} \} } \left( \varphi (g_p \cdots g_1 x) - \varphi (g_p \cdots g_1 x')\right) \nu(dx') \right| \nu(dx) \\ &\leqslant C e^{-a \alpha p} + 2 \|\varphi\|_{\infty} \int_{\mathbb{P}(\mathbb{V})} \mathbb{E} \; \nu \left\{ x' \in \mathbb{P}(\mathbb{V}): d(g_p \cdots g_1 x, g_p \cdots g_1 x') > e^{ - ap } \right\} \nu(dx) \\ &\leqslant C e^{-a \alpha p} + 2 \|\varphi\|_{\infty} e^{-b p} \\ & \quad+ 2 \|\varphi\|_{\infty} \int_{\mathbb{P}(\mathbb{V})} \mathbb{P} \left( \nu \left\{ x' \in \mathbb{P}(\mathbb{V}): d(g_p \cdots g_1 x, g_p \cdots g_1 x') > e^{ - ap } \right\} > e^{ - bp } \right) \nu(dx) \\ &\leqslant C e^{-a \alpha p} + 2 \|\varphi\|_{\infty} e^{-b p} + C e^{-cp}, \end{align}\] where in the last inequality we apply Lemma 1 with \(\varrho = \nu\). This proves 28 .

In the same way, for \(m \geqslant p\) and any \(x \in \mathbb{P}(\mathbb{V})\), we have, \(\mathbb{P}\)-almost surely, \[\begin{align} \mathbb{E} (\theta^{x, m} | \mathscr{A}_p) = \int_{\mathbb{G}^{m-p}} \varphi(g_1 \cdots g_p g'_{p+1} \cdots g'_{m} x) \mu(dg'_{p+1}) \ldots \mu(dg'_{m}), \end{align}\] which gives \[\begin{align} & \mathbb{E} | \theta^{x, m} - \mathbb{E} (\theta^{x, m} | \mathscr{A}_p) | \notag\\ & =\int_{\mathbb{G}^m} \left| \varphi (g_1 \cdots g_m x) - \int_{\mathbb{G}^{m-p}} \varphi (g_1 \cdots g_p g'_{p+1} \cdots g'_{m} x) \mu(dg'_{p+1}) \ldots \mu(dg'_{m}) \right| \notag\\ &\qquad\qquad \mu(dg_{1}) \ldots \mu(dg_{m}). \end{align}\] Following the same proof as above, we get 29 . ◻

Proof of Proposition 6. By 23 and 26 , for any \(y \in \mathbb{P}(\mathbb{V}^*)\), we have, \(\mathbb{P}\)-almost surely, \[\begin{align} \mathbb{E} (f(\cdot, y) | \mathscr{A}_p) = \int_{\mathbb{P}(\mathbb{V})} \delta(g_1 \cdots g_p x', y) \nu(dx'). \end{align}\] As mentioned before, by 27 , there exists a constant \(\alpha_0 > 0\) such that \[\begin{align} \label{new32expmoment001} \sup_{y \in \mathbb{P}(\mathbb{V}^*)} \int_{\mathbb{P}(\mathbb{V})} e^{ \alpha_0 \delta(x', y) } \nu(dx') < \infty. \end{align}\tag{30}\] This yields that, for any \(\alpha \in (0,\alpha_0/3]\), \[\begin{align} \label{eq-projappr-001} \int_{\Omega} e^{\alpha |f(\omega, y) - \mathbb{E} (f(\cdot, y) | \mathscr{A}_p)(\omega) |} \mathbb{P}(d \omega) = \mathbb{E} \left( \int_{\mathbb{P}(\mathbb{V})} \exp \left( \alpha \left| \bar{\delta}(G_px, y) \right| \right) \nu(dx) \right), \end{align}\tag{31}\] where, for short, we denote \(G_p = g_p \cdots g_1\) and \[\begin{align} \bar{\delta}(G_px, y) = \delta(G_px, y) - \int_{\mathbb{P}(\mathbb{V})} \delta(G_px', y) \nu(dx'). \end{align}\] Note that, by Hölder’s inequality and 30 , the expectation on the right-hand side of 31 can be shown to be finite. Therefore, by using Fubini’s theorem, we get, for any \(\alpha \in (0,\alpha_0/3]\), \[\begin{align} \int_{\Omega} e^{\alpha |f(\omega, y) - \mathbb{E} (f(\cdot, y) | \mathscr{A}_p)(\omega) |} \mathbb{P}(d \omega) = \int_{\mathbb{P}(\mathbb{V})} \mathbb{E} \exp \left( \alpha \left| \bar{\delta}(G_px, y) \right| \right) \nu(dx). \end{align}\] We fix \(\varepsilon\in (0, a/2)\), where \(a\) is as in Lemma 1. Set \(A_{p,x,y} = \{ \delta(G_px, y) > \varepsilon p \}\). By [11], we have uniformly in \(x \in \mathbb{P}(\mathbb{V})\) and \(y \in \mathbb{P}(\mathbb{V}^*)\), \[\begin{align} \label{exp32bound32Apx-001} \mathbb{P} \left( A_{p, x, y} \right) \leqslant C e^{ - cp }. \end{align}\tag{32}\] On the set \(A_{p, x, y}^c\), we have \[\begin{align} \left| \bar{\delta}(G_px, y) \right| & \leqslant\left| \int_{ B_{p, x}(\omega) } \left( \delta(G_px, y) - \delta(G_px', y) \right) \nu(dx') \right| \notag\\ & \quad + \left| \int_{ B_{p, x}(\omega)^c } \left( \delta(G_px, y) - \delta(G_px', y) \right) \nu(dx') \right|, \end{align}\] where \(B_{p, x}(\omega)\) is the random set \(\{x' \in \mathbb{P}(\mathbb{V}): d(G_px, G_px') \leqslant e^{-ap} \}\). Here and in the rest of this proof, we omit \(\omega\) in \(G_p(\omega)\) for short. On \(A_{p,x,y}^c\), for \(x' \in B_{p, x}(\omega)\), we have, by the mean value theorem, \[\begin{align} |\delta( G_px, y ) - \delta( G_px', y ) | \leqslant C e^{\delta (G_px, y)} e^{- ap} \leqslant C e^{ \varepsilon p } e^{- ap} \leqslant C e^{- \frac{a}{2} p}. \end{align}\] Hence, we obtain that on \(A_{p,x,y}^c\), \[\begin{align} \left| \bar{\delta}(G_px, y) \right| & \leqslant C e^{- \frac{a}{2} p} + \left| \int_{ B_{p, x}(\omega)^c } \left( \delta(G_px, y) - \delta(G_px', y) \right) \nu(dx') \right| \notag\\ & \leqslant C e^{- \frac{a}{2} p} + \left( \nu \left( B_{p, x}(\omega)^c \right) \right)^{1/2} \left( \int_{ \mathbb{P}(\mathbb{V}) } \left( \delta(G_px, y) - \delta(G_px', y) \right)^2 \nu(dx') \right)^{1/2} \notag\\ & \leqslant C e^{- \frac{a}{2} p} + \left( \nu \left( B_{p, x}(\omega)^c \right) \right)^{1/2} \left[ \delta(G_px, y) + \left( \int_{ \mathbb{P}(\mathbb{V}) } \delta(G_px', y)^2 \nu(dx') \right)^{1/2} \right] \notag\\ & \leqslant C e^{- \frac{a}{2} p} + \left( \nu \left( B_{p, x}(\omega)^c \right) \right)^{1/2} \left[ \varepsilon p + \left( \int_{ \mathbb{P}(\mathbb{V}) } \delta(G_px', y)^2 \nu(dx') \right)^{1/2} \right]. \end{align}\] We set \(E_{p,x} = \{ \omega \in \Omega: \nu ( B_{p, x}(\omega)^c ) > e^{ -bp } \}\). By Lemma 1, we have that \[\begin{align} \label{exp32bound32Epx-001} \mathbb{P} ( E_{p,x} ) \leqslant Ce^{-cp}. \end{align}\tag{33}\] Therefore, for any \(\alpha \in (0,\alpha_0/3]\), \[\begin{align} & \mathbb{E} \exp \left\{ \alpha \left| \bar{\delta}(G_px, y) \right| \right\} \leqslant\mathbb{E} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} } \exp \left\{ \alpha \left| \bar{\delta}(G_px, y) \right| \right\} \notag\\ & \quad + \mathbb{E} \mathbb{1}_{ E^c_{p,x} \cap A^c_{p,x} } \exp \left( \alpha C e^{- \frac{a}{2} p} + \alpha e^{- b p/2} \left[ \varepsilon p + \left( \int_{ \mathbb{P}(\mathbb{V}) } \delta(G_px', y)^2 \nu(dx') \right)^{1/2} \right] \right) . \end{align}\] As the measure \(\nu\) is \(\mu\)-stationary, we get \[\begin{align} \mathbb{E} \int_{ \mathbb{P}(\mathbb{V}) } \delta(G_px', y)^2 \nu(dx') = \int_{ \mathbb{P}(\mathbb{V}) } \delta(x', y)^2 \nu(dx') \leqslant C, \end{align}\] where \(C\) does not depend on \(y \in \mathbb{P}(\mathbb{V}^*)\) due to 30 . Set \[\begin{align} F_{p,y} = \left\{ \int_{ \mathbb{P}(\mathbb{V}) } \delta(G_px', y)^2 \nu(dx') > e^{\frac{bp}{2}} \right\}. \end{align}\] By Chebyshev’s inequality, we obtain \[\begin{align} \label{exp32bound32Fp-001} \mathbb{P}\left( F_{p,y} \right) \leqslant C e^{- \frac{bp}{2}}. \end{align}\tag{34}\] Now we get, for any \(\alpha \in (0,\alpha_0/3]\), \[\begin{align} & \mathbb{E} \exp \left\{ \alpha \left| \bar{\delta}(G_px, y) \right| \right\} \notag\\ & \leqslant\mathbb{E} \mathbb{1}_{ F_{p,y}^c \cap E^c_{p,x} \cap A^c_{p,x,y} } \exp \left( \alpha C e^{- \frac{a}{2} p} + \alpha e^{- b p/2} \left\{ \varepsilon p + e^{bp /4} \right\} \right) \notag\\ & \quad + \mathbb{E} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} \cup F_{p,y} } \exp \left( \alpha \left| \bar{\delta}(G_px, y) \right| \right) \notag \\ & \leqslant\exp \left( \alpha C e^{- \frac{a}{2} p} + \alpha e^{- b p/2} \left\{ \varepsilon p + e^{bp /4} \right\} \right) \notag\\ & \quad + \mathbb{E} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} \cup F_{p,y} } \exp \left( \alpha \left| \delta(G_px, y) \right| + \alpha \left| \int_{\mathbb{P}(\mathbb{V})} \delta(G_px', y) \nu(dx') \right| \right). \end{align}\] As \(p \to \infty\), the first term is bounded by \(1+e^{-cp}\), for some constant \(c>0\). We claim that if \(\alpha>0\) is chosen small enough, then the expectation of the second term tends to \(0\) at an exponential rate. Indeed, by Hölder’s inequality, we have, for any \(\alpha \in (0,\alpha_0/3]\), \[\begin{align} \label{Final32bound32projection-001001} &\int_{\mathbb{P}(\mathbb{V})} \mathbb{E} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} \cup F_{p,y} } \exp \left( \alpha \left| \delta(G_px, y) \right| + \alpha \left| \int_{\mathbb{P}(\mathbb{V})} \delta(G_px', y) \nu(dx') \right| \right) \nu(dx) \notag \\ &\leqslant \left( \int_{\mathbb{P}(\mathbb{V})} \mathbb{P} ( E_{p,x} \cup A_{p,x,y} \cup F_{p,y} ) \nu(dx)\right)^{1/3} \left(\int_{\mathbb{P}(\mathbb{V})} \mathbb{E} \exp \left( 3 \alpha \left| \delta(G_px, y) \right| \right) \nu(dx)\right)^{1/3} \notag \\ &\qquad\qquad\times \left[ \mathbb{E} \exp \left( 3 \alpha \left| \int_{\mathbb{P}(\mathbb{V})} \delta(G_px', y) \nu(dx') \right| \right) \right]^{1/3}. \end{align}\tag{35}\] From 32 , 33 and 34 , we have that, for some constant \(c>0\), uniformly in \(x\in \mathbb{P}(\mathbb{V})\), \(y \in \mathbb{P}(\mathbb{V}^*)\) and \(p\) large enough, \[\begin{align} \mathbb{P} ( E_{p, x} \cup A_{p, x, y} \cup F_{p, y} ) \leqslant e^{-cp}. \end{align}\] By Jensen’s inequality, the third factor on the right-hand side of 35 is less than the second factor. As the measure \(\nu\) is \(\mu\)-stationary, the latter is equal to \[\begin{align} \int_{\mathbb{P}(\mathbb{V})} \exp \Big( 3 \alpha \left| \delta(x, y) \right| \Big) \nu(dx), \end{align}\] which is finite by 30 , as soon as \(3\alpha \leqslant\alpha_0\). ◻

Proof. Our proof follows a similar approach as the one used in Proposition 6. Note that, by 36 , whenever \(p \geqslant m-n\), we have, for \(\mathbb{P}\)-almost all \(\omega \in \Omega\), \[\begin{align} f^{x,m}_{n}(\omega, y) - \mathbb{E} (f^{x,m}_{n}(\cdot, y) | \mathscr{A}_p)(\omega) = 0, \end{align}\] so the assertion of the proposition becomes evident. Therefore, in the remainder of the proof, we can assume that \(p < m-n\). For fixed \(x \in \mathbb{P}(\mathbb{V})\), still by 36 , we have, for \(\nu^*\)-almost every \(y \in \mathbb{P}(\mathbb{V}^*)\) and \(\mathbb{P}\)-almost every \(\omega=(g_1,g_2,\ldots) \in \Omega\), \[\begin{align} \mathbb{E} (f^{x,m}_{n}(\cdot, y) | \mathscr{A}_p)(\omega) = \int_{\mathbb{G}^{m-n-p}} \delta \Big(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y \Big) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}'). \end{align}\] For short, set \[\begin{align} \label{def-tilde-delta} \widetilde{\delta}(\omega, x, y) & = \delta(g_1 \cdots g_{m-n} x, y) \notag\\ & \quad - \int_{\mathbb{G}^{m-n-p}} \delta \Big( g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y \Big) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}'). \end{align}\tag{38}\] As before, by 27 , there exists a constant \(\alpha_0 > 0\) satisfying 30 . Applying Fubini’s theorem, for \(\alpha\in (0,\alpha_0/3]\), this yields \[\begin{align} & \int_{\mathbb{P}(\mathbb{V}^*)} \int_{\Omega} e^{\alpha |f^{x,m}_{n}(\omega, y) - \mathbb{E} (f^{x,m}_{n}(\cdot, y) | \mathscr{A}_p)(\omega) |} \mathbb{P}(d\omega) \nu^*(dy) \notag\\ & = \int_{\Omega} \int_{\mathbb{P}(\mathbb{V}^*)} \exp \left\{ \alpha \left| \widetilde{\delta}(\omega, x, y) \right| \right\} \nu^*(dy) \mathbb{P}(d\omega) \notag\\ & = \int_{\Omega} \int_{\mathbb{P}(\mathbb{V}^*)} \exp \bigg\{ \alpha \bigg| \int_{\mathbb{G}^{m-n-p}} \left( \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right) \notag\\ & \qquad \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \bigg| \bigg\} \nu^*(dy) \mathbb{P}(d\omega), \end{align}\] where the finiteness of the integrals is guaranteed by 30 . Let \(\varepsilon> 0\), whose value will be determined later. For \(y \in \mathbb{P}(\mathbb{V}^*)\), define the set \(A_{p,x,y} = \{ \omega \in \Omega: \delta (g_1 \cdots g_{m-n} x, y) > \varepsilon p \}\). Since \(p < m-n\), by [11], we have uniformly in \(x \in \mathbb{P}(\mathbb{V})\) and \(y \in \mathbb{P}(\mathbb{V}^*)\), \[\begin{align} \label{exp32bound32Apx-001-modi} \mathbb{P} \left( A_{p,x,y} \right) \leqslant C e^{ - cp }. \end{align}\tag{39}\] On the set \(A_{p,x,y}^c\), we have \[\begin{align} & \left| \int_{\mathbb{G}^{m-n-p}} \left( \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right| \notag\\ & \leqslant\left| \int_{ B_{p,x}(\omega) } \left( \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right| \notag\\ & \quad + \left| \int_{ B_{p,x}(\omega)^c } \left( \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right|, \end{align}\] where \(B_{p,x}(\omega)\) is the random set \[\begin{align} \Big\{ (g_{p+1}', \cdots, g_{m-n}') \in \mathbb{G}^{m-n-p}: d \Big( g_1 \cdots g_{m-n} x, g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x \Big) &\leqslant e^{-ap} \Big\} \\ &\subseteq \mathbb{G}^{m-n-p}. \end{align}\] By Lemma 1, we have \[\begin{align} \label{inequa-random-measure-modified} \mathbb{P} \left( \mu^{\otimes (m-n-p)} \left( B_{p,x}(\omega) \right) > e^{ - bp } \right) \leqslant C e^{-cp}. \end{align}\tag{40}\] On \(A_{p,x,y}^c\), for \((g_{p+1}', \ldots, g_{m-n}') \in B_{p, x}(\omega)\), we get, by the mean value theorem, \[\begin{align} & \left| \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right| \notag\\ & \leqslant C e^{\delta (g_1 \cdots g_{m-n} x, y)} e^{- ap} \leqslant C e^{ \varepsilon p } e^{- ap} \leqslant C e^{- \frac{a}{2} p}, \end{align}\] where we have assumed that \(\varepsilon\leqslant a/2\). Hence, we obtain that for any \(\omega \in A_{p,x,y}^c\), \[\begin{align} & \left| \int_{\mathbb{G}^{m-n-p}} \left( \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right| \notag\\ & \leqslant C e^{- \frac{a}{2} p} \\ & \quad + \left| \int_{ B_{p,x}(\omega)^c } \left( \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right| \notag\\ & \leqslant C e^{- \frac{a}{2} p} + \left( \mu^{\otimes (m-n-p)} \left( B_{p, x}(\omega)^c \right) \right)^{1/2} \notag\\ & \quad \times \left[ \int_{ \mathbb{G}^{m-n-p} } \left( \delta(g_1 \cdots g_{m-n} x, y) - \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \right)^2 \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right]^{1/2} \notag\\ & \leqslant C e^{- \frac{a}{2} p} + \left( \mu^{\otimes (m-n-p)} \left( B_{p, x}(\omega)^c \right) \right)^{1/2} \notag\\ & \quad \times \left\{ \delta(g_1 \cdots g_{m-n} x, y) + \left[ \int_{ \mathbb{G}^{m-n-p} } \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y)^2 \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right]^{1/2} \right\} \notag\\ & \leqslant C e^{- \frac{a}{2} p} + \left( \mu^{\otimes (m-n-p)} \left( B_{p, x}(\omega)^c \right) \right)^{1/2} \notag\\ & \quad \times \left\{ \varepsilon p + \left[ \int_{ \mathbb{G}^{m-n-p} } \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y)^2 \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right]^{1/2} \right\}. \end{align}\] We set \(E_{p,x} = \{ \mu^{\otimes (m-n-p)} ( B_{p, x}(\omega)^c ) > e^{ -bp } \}\). By 40 , we have \[\begin{align} \label{exp32bound32Epx-001-modi} \mathbb{P} ( E_{p,x} ) \leqslant Ce^{-cp}. \end{align}\tag{41}\] Therefore, for \(\alpha\in (0,\alpha_0/3]\), \[\begin{align} & \int_{\mathbb{P}(\mathbb{V}^*)} \int_{\Omega} e^{\alpha |f^{x,m}_{n}(\omega, y) - \mathbb{E} (f^{x,m}_{n}(\cdot, y) | \mathscr{A}_p)(\omega) |} \mathbb{P}(d\omega) \nu^*(dy) \notag\\ & \leqslant\int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{E} \mathbb{1}_{ E^c_{p,x} \cap A^c_{p,x,y} } \exp \Bigg( \alpha C e^{- \frac{a}{2} p} + \alpha e^{- b p/2} \Bigg\{ \varepsilon p \notag\\ & \quad + \bigg[ \int_{ \mathbb{G}^{m-n-p} } \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y)^2 \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \bigg]^{1/2} \Bigg\} \Bigg) \nu^*(dy) \notag\\ & \quad + \int_{\mathbb{P}(\mathbb{V}^*)} \int_{\Omega} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} } \exp \left\{ \alpha \left| \widetilde{\delta}( \omega, x, y) \right| \right\} \mathbb{P}(d \omega) \nu^*(dy), \end{align}\] where \(\widetilde{\delta}(\omega, x, y)\) is defined by 38 . Now we have \[\begin{align} \label{Triw-modified-inequa} & \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{E} \int_{ \mathbb{G}^{m-n-p} } \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y)^2 \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \nu^*(dy) \notag\\ & \leqslant\sup_{x' \in \mathbb{P}(\mathbb{V})} \int_{\mathbb{P}(\mathbb{V}^*)} \delta(x', y)^2 \nu^*(dy) \leqslant C, \end{align}\tag{42}\] where the last inequality holds due to [11]. For \(y \in \mathbb{P}(\mathbb{V}^*)\), set \[\begin{align} F_{p,y} = \left\{ \omega \in \Omega: \int_{ \mathbb{G}^{m-n-p} } \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y)^2 \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') > e^{\frac{bp}{2}} \right\}. \end{align}\] By Chebyshev’s inequality and 42 , we obtain \[\begin{align} \label{exp32bound32Fp-001-modi} \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{P} \left( F_{p,y} \right) \nu^*(dy) \leqslant C e^{- \frac{bp}{2}}. \end{align}\tag{43}\] Then, we derive that, for \(\alpha\in (0,\alpha_0/3]\), \[\begin{align} & \int_{\mathbb{P}(\mathbb{V}^*)} \int_{\Omega} e^{\alpha |f^{x,m}_{n}(\omega, y) - \mathbb{E} (f^{x,m}_{n}(\omega, y) | \mathscr{A}_p) |} \mathbb{P}(d\omega) \nu^*(dy) \notag\\ & \leqslant\int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{E} \mathbb{1}_{ F_{p,y}^c \cap E^c_{p,x} \cap A^c_{p,x,y} } \exp \left( \alpha C e^{- \frac{a}{2} p} + \alpha e^{- b p/2} \left\{ \varepsilon p + e^{bp /4} \right\} \right) \nu^*(dy) \notag\\ & \quad + \int_{\mathbb{P}(\mathbb{V}^*)} \int_{\Omega} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} \cup F_{p,y} } \exp \left\{ \alpha \left| \widetilde{\delta}(\omega, x, y) \right| \right\} \mathbb{P}(d \omega) \nu^*(dy) \notag \\ & \leqslant\exp \left( \alpha C e^{- \frac{a}{2} p} + \alpha e^{- b p/2} \left\{ \varepsilon p + e^{bp /4} \right\} \right) \notag\\ & \quad + \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{E} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} \cup F_{p,y} } \exp \Bigg\{ \alpha \left| \delta(g_1 \cdots g_{m-n} x, y) \right| \notag\\ & \qquad\qquad + \alpha \left| \int_{\mathbb{G}^{m-n-p}} \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right| \Bigg\} \nu^*(dy). \end{align}\] As \(p \to \infty\), the first term is bounded by \(1+e^{-cp}\), for some constant \(c>0\). We claim that if \(\alpha>0\) is chosen small enough, then the expectation of the second term goes to \(0\) with an exponential rate. Indeed, by Hölder’s inequality, we have \[\begin{align} \label{Final32bound32projection-001001-modi} & \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{E} \mathbb{1}_{ E_{p,x} \cup A_{p,x,y} \cup F_{p,y} } \exp \Bigg\{ \alpha \left| \delta(g_1 \cdots g_{m-n} x, y) \right| \notag\\ & \qquad\qquad\qquad + \alpha \left| \int_{\mathbb{G}^{m-n-p}} \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \right| \Bigg\} \nu^*(dy) \notag \\ &\leqslant \left( \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{P} ( E_{p,x} \cup A_{p,x,y} \cup F_{p,y} ) \nu^*(dy)\right)^{1/3} \notag\\ & \quad \times \left( \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{E} \exp \left\{ 3 \alpha \left| \delta(g_1 \cdots g_{m-n} x, y) \right| \right\} \nu^*(dy) \right)^{1/3} \notag \\ &\quad \times \Big( \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{E} \exp \Big\{ 3 \alpha \Big| \int_{\mathbb{G}^{m-n-p}} \delta(g_1 \cdots g_p g_{p+1}' \cdots g_{m-n}' x, y) \notag \\ & \qquad\qquad \mu(dg_{p+1}') \ldots \mu(dg_{m-n}') \Big| \Big\} \nu^*(dy) \Big)^{1/3}. \end{align}\tag{44}\] From 39 , 41 and 43 , we have that, for some \(c>0\), uniformly in \(x\in \mathbb{P}(\mathbb{V})\) and \(p\) large enough, \[\begin{align} \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{P} ( E_{p,x} \cup A_{p,x,y} \cup F_{p,y} ) \nu^*(dy) \leqslant e^{-cp}. \end{align}\] By Jensen’s inequality, the third factor on the right-hand side of 44 is less than the second factor. The latter is bounded by \[\begin{align} \sup_{x' \in \mathbb{P}(\mathbb{V})} \int_{\mathbb{P}(\mathbb{V}^*)} \exp \left\{ 3 \alpha \left| \delta(x', y) \right| \right\} \nu^*(dy), \end{align}\] which, by 30 , is finite as soon as \(3\alpha\leqslant\alpha_0\). The conclusion of the proposition follows. ◻

Proof. First assume that the function \(\varphi\) is Hölder continuous on \(\mathbb{P}(\mathbb{V})\). By Corollary 5, the function \(\omega \mapsto \varphi(\xi(\omega))\) satisfies the approximation property 84 . Therefore, by Corollary 7, for any non-negative continuous compactly supported function \(\psi\) on \(\mathbb{R}\), as \(n\to \infty\), the quantity \(\int_{\mathbb{R}} U^{\varphi}_{n}(t) \psi(t) dt\) has a limit of the form \(\int_{\mathbb{R}} U^{\varphi}(t) \psi(t) dt\) for some non-decreasing function \(U^{\varphi}: \mathbb{R} \to \mathbb{R}_+\).

In the case where \(\varphi\) is any continuous function on \(\mathbb{P}(\mathbb{V})\), for any \(\varepsilon>0\), we can choose a Hölder continuous function \(\varphi'\) on \(\mathbb{P}(\mathbb{V})\) such that \(\sup_{x \in \mathbb{P}(\mathbb{V})} |\varphi(x)-\varphi'(x)| \leqslant\varepsilon\). Then, for any \(\psi\) as above and any \(n\geqslant 1\), we have, using Corollary 8, \[\begin{align} \left| \int_{\mathbb{R}} U^{\varphi}_{n}(t) \psi(t) dt - \int_{\mathbb{R}} U^{\varphi'}_{n}(t) \psi(t) dt \right| &\leqslant\varepsilon\int_{\mathbb{R}} U_{n}(t) \psi(t) dt \\ &\leqslant c \varepsilon\int_{\mathbb{R}} (1+\max{t,0}) \psi(t) dt. \end{align}\] Since, by the first part of the proof, the integral \(\int_{\mathbb{R}} U^{\varphi'}_{n}(t) \psi(t) dt\) has a limit, the inequality above shows that the sequence of integrals \(\int_{\mathbb{R}} U^{\varphi}_{n}(t) \psi(t) dt,\) \(n\geqslant 1\), forms a Cauchy sequence and thus also converges as \(n\to\infty\). In other words, for any continuous compactly supported function \(h\) on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\) which is of the product form \((x, t) \mapsto \varphi(x) \psi(t)\), the integral \[\begin{align} \label{INT-EXPECT-001} \int_{\mathbb{P}(\mathbb{V}^*) \times \mathbb{R}} \int_{\Omega} &\bigg[ \bigg( t + \tilde{S}_n(\omega,y) \bigg) h(\xi(\omega) , t); \tilde{\tau}_{y,t}(\omega) >n \bigg] \mathbb{P}(d\omega) \nu^*(dy) dt \end{align}\tag{48}\] has a limit as \(n\to\infty\). Since every continuous compactly supported function \(h\) on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\) can be uniformly approximated by sums of functions of the product form, the same argument as above shows that the integral 48 has a limit for any \(h\). As this integral is non-negative when \(h\) is non-negative, by the Riesz representation theorem, the limit is of the form \(\rho (h)\), where \(\rho\) is a Radon measure on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\).

The properties of the marginal of the measure \(\rho\) on \(\mathbb{R}\) are direct consequences of Corollaries 7 and 8. Finally, we prove ?? . Assume first that \(h\) is a continuous compactly supported function on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\) which is of the form \((x, t) \mapsto \varphi(x) \psi(t)\), where \(\varphi\) is a Hölder continuous function on \(\mathbb{P}(\mathbb{V})\) and \(\psi\) is a continuous compactly supported function on \(\mathbb{R}\). Then, by Corollaries 5 and 7, we obtain \[\begin{align} \lim_{t \to \infty} \frac{1}{t} \int_{\mathbb{R}} U^{\varphi}(t') \psi(t' - t) dt' &= \int_{\mathbb{R}} \psi(t') dt' \int_{\Omega} \varphi(\xi(\omega)) \mathbb{P}(d \omega) \\ &= \int_{\mathbb{R}} \psi(t') dt' \int_{\mathbb{P}(\mathbb{V})} \varphi(x) \nu(dx), \end{align}\] which establishes ?? when \(h\) is of the above form.

The general case of ?? follows by standard approximation and by using the bound ?? . Indeed, if \(h\) is any continuous compactly supported function on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\), then for any \(\varepsilon> 0\), we can find an integer \(m \geqslant 1\), Hölder continuous functions \(\varphi_1, \ldots, \varphi_m\) on \(\mathbb{P}(\mathbb{V})\) and continuous compactly supported functions \(\psi_1, \ldots, \psi_m\) on \(\mathbb{R}\), such that \[\begin{align} \label{approxim32by32finite32sum32of32prod-001} \sup_{(x, t) \in \mathbb{P}(\mathbb{V}) \times \mathbb{R}} \Big| h(x, t) - \sum_{i = 1}^m \varphi_i(x) \psi_i(t) \Big| \leqslant\varepsilon. \end{align}\tag{49}\] Thus, for any \(t \in \mathbb{R}\), we get \[\begin{align} & \left| \int_{ \mathbb{P}(\mathbb{V}) \times \mathbb{R} } h(x', t' - t) \rho(dx', dt') - \sum_{i = 1}^m \int_{ \mathbb{P}(\mathbb{V}) \times \mathbb{R} } \varphi_i(x') \psi_i(t' - t) \rho(dx', dt') \right| \notag\\ & \leqslant\varepsilon\rho(\mathbb{P}(\mathbb{V}) \times [t-C, t + C] ), \end{align}\] for some constant \(C>0\) depending on the support of the above functions. Using the bound ?? , we get \(\rho(\mathbb{P}(\mathbb{V}) \times [t-C, t + C] ) \leqslant c (t + 2C).\) Dividing by \(t\) and using ?? which has been already established for functions \(h\) of the form \((x, t) \mapsto \varphi(x) \psi(t)\), we obtain \[\begin{align} \limsup_{t \to \infty} \left| \frac{1}{t} \int_{ \mathbb{P}(\mathbb{V}) \times \mathbb{R} } h(x', t' - t) \rho(dx', dt') - \sum_{i = 1}^m \int_{ \mathbb{P}(\mathbb{V}) \times \mathbb{R} } \varphi_i(x') \psi_i(t') \nu(dx') dt' \right| \leqslant c \varepsilon. \end{align}\] The result ensues from the application of 49 and letting \(\varepsilon\to 0\). ◻

Proof. Fix \(x\in \mathbb{P}(\mathbb{V})\). By the definition of \(\delta\) (cf.@eq:def32of32delta32func-001 ) and 25 , for any \(y \in \mathbb{P}(\mathbb{V}^*)\) and \(g \in \mathbb{G}\), we have that \(\delta(gx, y) = \infty\) if and only if \(\delta(x, g^{-1} y) = \infty\). Therefore, in order to establish the lemma, it is equivalent to prove that, \(\nu^*\)-almost surely in \(y\in \mathbb{P}(\mathbb{V}^*)\), for any \(n\geqslant 1\), \(\mathbb{P} \left( \delta(x, (g_n \cdots g_1)^{-1} y) < \infty \right) = 1\).

As the measure \(\nu^*\) assigns zero mass to every proper projective subspace of \(\mathbb{P}(\mathbb{V}^*)\) (see Lemma 4.6 in [11]), we have \[\begin{align} \nu^*\big( y\in \mathbb{P}(\mathbb{V}^*): \delta(x,y) = \infty \big) =0. \end{align}\] Since \(\nu^*\) is \(\mu^{-1}\)-stationary, we get that for any \(n\geqslant 1\), \[\begin{align} & \int_{\mathbb{P}(\mathbb{V}^*)} \mathbb{P} \left( \delta(x, (g_n \cdots g_1)^{-1} y) = \infty \right) \nu^*(dy) \notag\\ & = \mathbb{E} \left[ \nu^*\left( y\in \mathbb{P}(\mathbb{V}^*): \delta(x, (g_n \cdots g_1)^{-1} y) = \infty \right) \right] \notag\\ & = \nu^*\left( y\in \mathbb{P}(\mathbb{V}^*): \delta(x,y) = \infty \right) = 0, \end{align}\] which ends the proof of the lemma. ◻

Proof. For brevity, set \(S^x_n= \sigma(g_n\cdots g_1, x)\) for \(n\geqslant 1\) and \(S^x_0=0\). Using the change of variable \(t = u+ S^x_n\), the left-hand side of 53 can be rewritten as \[\begin{align} \label{After32a32change32of32variable-001} J: & =\int_{\mathbb{R}_+} u \mathbb{E} \Big[ h \Big( g_n\cdots g_1 x, u+ S^x_n \Big); \tau_{x,u} > n-1 \Big] du \notag\\ & = \int_{\mathbb{R}} \mathbb{E} \Big[ u h \Big(g_n\cdots g_1 x, u+ S^x_n\Big); u + S^x_k \geqslant 0, 0\leqslant k\leqslant n-1 \Big] du \notag\\ & = \int_{\mathbb{R}} \mathbb{E}\Big[ \Big(t - S^x_n \Big) h(g_n\cdots g_1 x, t); t - S^x_n + S^x_k \geqslant 0, 0\leqslant k\leqslant n-1 \Big] dt. \end{align}\tag{54}\] In the last integral, we have, for \(0\leqslant k\leqslant n-1\), \[\begin{align} S^x_n - S^x_k = \sigma(g_{n}\cdots g_{k+1}, g_{k}\cdots g_{1} x ), \end{align}\] where by convention \(g_{k}\cdots g_{1} x=x\) for \(k=0\). We shall make use of the following cohomological equation, which is obtained from 25 : for \(g \in \mathbb{G}\), \(a \in \mathbb{P}(\mathbb{V})\) and \(b \in \mathbb{P}(\mathbb{V}^*)\), we have \[\begin{align} \label{32cohomological32eq32-vers32003} \sigma(g, a) = \sigma^*(g^{-1}, b) - \delta (a, g^{-1}b) + \delta(g a, b). \end{align}\tag{55}\] Then, with \(g =g_{n}\cdots g_{k+1}\), \(a =g_{k}\cdots g_{1} x\) and \(b=y\), we get, for \(0\leqslant k\leqslant n\), \[\begin{align} \label{32cohomological32eq32-vers32004} &S^x_n - S^x_k = \sigma(g_{n}\cdots g_{k+1}, g_{k}\cdots g_{1} x ) \notag \\ &= \sigma^* \left( (g_{n}\cdots g_{k+1})^{-1}, y \right) - \delta \left( g_{k}\cdots g_{1} x, (g_{n}\cdots g_{k+1})^{-1}y \right) + \delta(g_{n}\cdots g_{1} x, y). \end{align}\tag{56}\] Using identity 56 we have \[\begin{align} J&= \int_{\mathbb{R}} \mathbb{E}\Big[ \Big(t -\sigma^*( g^{-1}_1\cdots g^{-1}_n,y) + \delta(x,g^{-1}_1\cdots g^{-1}_{n}y)-\delta(g_n\cdots g_1x,y) \Big) h(g_n\cdots g_1 x, t); \notag\\ &\qquad\qquad t -\sigma^*( g^{-1}_{k+1}\cdots g^{-1}_n,y) + \delta(g_k\cdots g_1 x, g^{-1}_{k+1} \cdots g^{-1}_{n}y)-\delta(g_n\cdots g_1x,y) \geqslant 0, \notag\\ & \qquad\qquad\qquad\qquad 0\leqslant k\leqslant n-1 \Big] dt. \end{align}\] The random elements \(g_1,\ldots,g_n\) are exchangeable, which is justified by the assumption that these elements are independent and identically distributed. Therefore, we can reverse the order of the elements \(g_1,\ldots,g_n\), which gives \[\begin{align} J & = \int_{\mathbb{R}} \mathbb{E}\Big[ \Big(t-\sigma^*( g^{-1}_n\cdots g^{-1}_1,y) + \delta(x,g^{-1}_n\cdots g^{-1}_{1}y) - \delta(g_1\cdots g_nx,y) \Big) h(g_1\cdots g_n x, t); \notag \\ & \qquad\qquad t-\sigma^*( g^{-1}_{k}\cdots g^{-1}_1,y) + \delta(g_{k+1}\cdots g_n x, g^{-1}_{k} \cdots g^{-1}_{1}y) - \delta(g_1\cdots g_nx,y) \geqslant 0, \notag\\ & \qquad\qquad\qquad\qquad 1\leqslant k\leqslant n \Big] dt \notag\\ & = \int_{\mathbb{R}} \mathbb{E}\Big[ \Big(t + \tilde{S}^{x,n}_{n}(\cdot, y) \Big) h(g_1\cdots g_n x, t); t + \tilde{S}^{x,n}_{k}(\cdot, y) \geqslant 0, 1\leqslant k\leqslant n \Big] dt. \end{align}\] This is exactly the integral on the right-hand side of 53 . ◻

Proof. This statement is a translation of Theorem 10 in the setting of products of random matrices. Indeed, we can check that the assumptions of this theorem are satisfied. First, we extend the sequence of perturbations \((f_{n}^{x,m})_{0 \leqslant n \leqslant m}\) as a full sequence \((f_{n}^{x,m})_{n \geqslant 0}\) as follows: for \(n > m\), \(\omega \in \Omega\) and \(y \in \mathbb{P}(\mathbb{V}^*)\), we set, for example, \(f_{n}^{x,m}(\omega, y)= f_{m}^{x,m}(\omega, y) = \delta(x, y)\). (Actually, the precise values of \(f_{n}^{x,m}\) for \(n > m\) will not play a role in our computations.) We also extend the random walk \((\tilde{S}^{x,m}_{n}(y))_{1\leqslant n \leqslant m}\) as a full random walk \((\tilde{S}^{x,m}_{n}(y))_{n \geqslant 1}\) by using the same formula as in 59 . Therefore, for any fixed \(x\in \mathbb{P}(\mathbb{V})\) and \(m\geqslant 1\), the random walk \((\tilde{S}^{x,m}_n(y))_{n\geqslant 1}\) has the form required in Theorem 10. As already mentioned, it follows from Lemma 10.18 in [11] that the cocycle \(\sigma^*\) can be centered, meaning that there exists a continuous function \(\psi_0\) on \(\mathbb{P}(\mathbb{V}^*)\) such that, for any \(y\in \mathbb{P}(\mathbb{V}^*)\), \[\begin{align} \int_{\mathbb{G} } \left( \sigma^*(g^{-1},y) + \psi_0(g^{-1} y)-\psi_0(y) \right) \mu(dg) = 0. \end{align}\] In other words, up to a bounded coboundary, the assumption 72 is satisfied with the cocycle \(\sigma = -\sigma^*.\) Besides, the effective central limit theorem 79 holds true, by the Berry-Esseen theorem in Bougerol-Lacroix [10] (Theorem 5.1, page 122). The perturbation sequence \(\mathfrak f =(f^{x,m}_n)_{n\geqslant 1}\) satisfies the bound 76 by virtue of inequality 37 , and possesses the approximation property 78 due to Proposition 7. Finally, the approximation property 84 is satisfied thanks to Corollary 5. ◻

Proof. Recall that for \(\mathbb{P}\)-almost every \(\omega = ( g_1, g_2, \ldots) \in \Omega\) and any \(n \geqslant 1\), we have \(\xi(\omega) = g_1 \cdots g_n \xi (T^n \omega)\). Therefore, by [11] (see equation (14.6)), we may find constants \(a, b, c>0\) such that for \(x \in \mathbb{P}(\mathbb{V})\) and \(n \geqslant 1\), \[\begin{align} \mathbb{P} \left(d \left( g_1 \cdots g_{n} x, \xi (\omega) \right) > e^{-an} \right) \leqslant ce^{-bn}. \end{align}\] Besides, still by [11], we may also assume that, for any \(x \in \mathbb{P}(\mathbb{V})\), \(y \in \mathbb{P}(\mathbb{V}^*)\) and \(n \geqslant 1\), \[\begin{align} \mathbb{P} \left( \delta\left( g_1 \cdots g_n x, y \right) > \frac{a}{2} n \right) \leqslant c e^{-bn}. \end{align}\] By the mean value theorem, we obtain, for any \(x \in \mathbb{P}(\mathbb{V})\), \(y \in \mathbb{P}(\mathbb{V}^*)\) and \(n \geqslant 1\), \[\begin{align} \mathbb{P} \left( \left| \delta (g_1 \cdots g_n x, y) - \delta (\xi (\omega), y) \right| > e^{-\frac{a}{2} n}, \delta (g_1 \cdots g_n x, y) < \infty \right) \leqslant 3c e^{-bn}, \end{align}\] which ends the proof of the lemma. ◻

Proof. We only provide a proof of the upper bound, as the lower bound can be obtained in a similar manner. Let \(a, b>0\) be as in Lemma 4. For \(x \in \mathbb{P}(\mathbb{V})\) and \(1 \leqslant n \leqslant m\), we set \[\begin{align} & E_{n}^{x, m} = \Big\{(\omega, y) \in \Omega \times \mathbb{P}(\mathbb{V}^*): d(g_1 \cdots g_m x, \xi(\omega)) \leqslant e^{-am}, \notag\\ & \quad \forall k \in [0, n], \left| f\circ \tilde{T}^k(\omega, y) - f_k^{x,m}\circ \tilde{T}^k(\omega, y) \right| \leqslant e^{-a(m-k)}, f_k^{x,m}\circ \tilde{T}^k(\omega, y) < \infty \Big\} \end{align}\] and let \((E_{n}^{x, m})^c\) be its complement. By Lemmas 1 and 4, we have \[\mathbb{P} \otimes \nu^* ((E_{n}^{x, m})^c) \leqslant c \sum_{k=0}^n e^{-b(m-k)} \leqslant c' e^{-b(m-n)}.\] Then using 60 , we obtain \[\begin{align} U^{x,m,\varphi}_{n}(t) &= \int_{\mathbb{P}(\mathbb{V}^*)} \int _{\Omega} \Big[ \Big( t + \tilde{S}^{x,m}_{n}(\omega,y)\Big) \varphi(g_1(\omega)\cdots g_m(\omega) x); \\ & \qquad\qquad\qquad t + \tilde{S}^{x,m}_{k}(\omega,y) \geqslant 0, 1\leqslant k\leqslant n, (\omega, y) \in E_{n}^{x, m} \Big] \mathbb{P}(d\omega) \nu^*(dy) \notag\\ & \quad + \int_{\mathbb{P}(\mathbb{V}^*)} \int _{\Omega} \Big[ \Big( t + \tilde{S}^{x,m}_{n}(\omega,y)\Big) \varphi(g_1(\omega)\cdots g_m(\omega) x); \\ &\qquad\qquad\qquad t + \tilde{S}^{x,m}_{k}(\omega,y) \geqslant 0, 1\leqslant k\leqslant n, (\omega, y) \notin E_{n}^{x, m} \Big] \mathbb{P}(d\omega) \nu^*(dy). \end{align}\] The first integral is dominated by \[\begin{align} & U_n^{\varphi}(t + 2 e^{-a(m-n)}) + Ce^{-cm} U_n^{\mathbf{1}}( t + 2 e^{-a(m-n)}) \notag\\ & \leqslant U_n^{\varphi}(t + 2 e^{-a(m-n)}) + Ce^{-cm} (1 + \max\{t, 0\}), \end{align}\] where in the last inequality we used Corollary 8. By the Cauchy-Schwarz inequality, Lemma 10 and inequality 37 , the second integral is bounded by \[\begin{align} \Big( \mathbb{P} \otimes \nu^* ((E_{n}^{x, m})^c) \Big)^{1/2} \left( \max\{t, 0\} + c \sqrt{n} \right) \leqslant c' e^{- \frac{b}{2}(m-n)} \left( \max\{t, 0\} + c \sqrt{n} \right), \end{align}\] which concludes the proof. ◻

Proof of Theorem 2. Using Proposition 9 and the fact that \(\psi\) is non-negative, there exist constants \(\beta, \gamma, c >0\) such that, for any \(x \in \mathbb{P}(\mathbb{V})\) and \(n \geqslant 1\), \[\begin{align} \label{ideal-003} \int_{\mathbb{R}} \psi(t) U_n^{x,n,\varphi}(t) dt \leqslant\int_{\mathbb{R}} \psi(t) U_{[n/2]}^{x,n,\varphi}(t + c n^{-\gamma}) dt + c n^{-\beta} \int_{\mathbb{R}} \psi(t) (1+\max \{t,0\}) dt \end{align}\tag{62}\] and \[\begin{align} \label{ideal-003bbb} \int_{\mathbb{R}} \psi(t) U_n^{x,n,\varphi}(t) dt \geqslant\int_{\mathbb{R}} \psi(t) U_{[n/2]}^{x,n,\varphi}(t - c n^{-\gamma}) dt - c n^{-\beta} \int_{\mathbb{R}} \psi(t) (1+\max \{t,0\}) dt. \end{align}\tag{63}\] By 62 and Corollary 6, there exist constants \(\beta, \gamma, c >0\) such that, for any \(x \in \mathbb{P}(\mathbb{V})\) and \(n \geqslant 1\), \[\begin{align} \label{ideal-004} \int_{\mathbb{R}} \psi(t) U_n^{x,n,\varphi}(t) dt & \leqslant\int_{\mathbb{R}} \psi(t) U_{[n/2]}^{\varphi} \left(t + c n^{-\gamma} \right) dt + c n^{-\beta} \int_{\mathbb{R}} \psi(t) (1+\max \{t,0\}) dt \notag\\ & = \int_{\mathbb{R}} \psi(t - c n^{-\gamma}) U^{\varphi}_{[n/2]} \left(t \right) dt + c n^{-\beta} \int_{\mathbb{R}} \psi(t) (1+\max \{t,0\}) dt \notag\\ & \leqslant\int_{\mathbb{R}} \psi(t) U^{\varphi}_{[n/2]} \left( t \right) dt + \int_{\mathbb{R}} \sup_{|u| \leqslant c n^{-\gamma}} | \psi(t + u) - \psi(t)| U^{\varphi}_{[n/2]} \left( t \right) dt \notag\\ & \quad + c n^{-\beta} \int_{\mathbb{R}} \psi(t) (1+\max \{t,0\}) dt. \end{align}\tag{64}\] By Proposition 8 and the continuity of \(\psi\), we obtain, uniformly in \(x \in \mathbb{P}(\mathbb{V})\), \[\begin{align} & \limsup_{n \to \infty} \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} h(x', t) \rho_{n,x}(dx', dt) = \limsup_{n \to \infty} \int_{\mathbb{R}} \psi(t) U_n^{x,n,\varphi}(t) dt \notag\\ & \qquad \leqslant\int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} \varphi(x') \psi(t) \rho(dx', dt) = \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} h(x', t) \rho(dx', dt). \end{align}\] Reasoning in the same way, and using 63 instead of 62 we get, uniformly in \(x \in \mathbb{P}(\mathbb{V})\), \[\begin{align} \liminf_{n \to \infty} \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} h(x', t) \rho_{n,x}(dx', dt) = \liminf_{n \to \infty} \int_{\mathbb{R}} \psi(t) U_n^{x,n,\varphi}(t) dt \geqslant\int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} h(x', t) \rho(dx', dt). \end{align}\] Therefore, the first assertion 10 follows for functions of the form \(h(x, t) = \varphi(x) \psi(t)\), where \(\varphi\) is a non-negative Hölder continuous on \(\mathbb{P}(\mathbb{V})\), and \(\psi\) is a compactly supported non-negative continuous on \(\mathbb{R}\). The extension to any continuous compactly supported function \(h\) on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\) follows by standard approximation arguments. The shows 10 . The proof of 11 can be done in the same way. ◻

Proof of Corollary 1. This is a consequence of Proposition 8. ◻

Proof of Corollary 2. Define a Borel measure \(\rho'\) on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\) by setting, for any non-negative Borel measurable function \(h\) on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\), \[\begin{align} \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} h(x,t) \rho'(dx,dt) = \int_{\mathbb{P}(\mathbb{V}) \times [0,\infty) } \mathbb{E} h \Big(g_1 x, t + \sigma(g_1, x) \Big) \rho(dx,dt). \end{align}\] We claim that \(\rho'\) is a Radon measure on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\), meaning that it is finite on compact sets. Indeed, since \(\mu\) has a finite exponential moment, by the bound ?? of Proposition 8, there exists a constant \(\alpha>0\) such that, for any \(a\geqslant 0\), \[\begin{align} \rho'(\mathbb{P}(\mathbb{V}) \times (-\infty,a]) & = \int_{\mathbb{P}(\mathbb{V}) \times [0,\infty) } \mathbb{P} \Big( t + \sigma(g_1, x) \leqslant a \Big) \rho(dx,dt) \\ & \leqslant c \int_{0}^{\infty} e^{-\alpha (t-a) } (1+t) dt <\infty. \end{align}\]

Now we show that the Radon measure \(\rho'\) coincides with the Radon measure \(\rho\) on continuous compactly supported functions. Indeed, fix \(x\in \mathbb{P}(\mathbb{V})\). For any continuous compactly supported function \(h\) on \(\mathbb{P}(\mathbb{V}) \times \mathbb{R}\), we have, by Theorem 2 and the Markov property, \[\begin{align} \label{intRPhhh000aa} & \int_{\mathbb{P}(\mathbb{V}) \times [0,\infty) } h (x',t') \rho(dx',dt') \notag \\ & = \lim_{n\to\infty} \int_{\mathbb{P}(\mathbb{V}) \times [0,\infty) } t \, \mathbb{E} \Big( h(g_n\cdots g_1 x, t + \sigma(g_n\cdots g_1, x)) ; \tau_{x,t} >n-1 \Big) dt \notag \\ & = \lim_{n\to\infty} \int_{\mathbb{P}(\mathbb{V}) \times [0,\infty) } t \, \mathbb{E} \Big( P h (g_{n-1}\cdots g_{1} x, t + \sigma(g_{n-1}\cdots g_1, x)) ; \tau_{x,t} >n-1 \Big) dt, \end{align}\tag{65}\] where we used the notation \[\begin{align} Ph(x',t') = \mathbb{E} h \Big( g_1 x',t'+\sigma(g_1,x') \Big). \end{align}\] Note that \(Ph\) is continuous, but we cannot apply directly Theorem 2, since the function \((x',t') \mapsto Ph(x',t') \mathbb{1}_{\{t'\geqslant 0 \}}\) may not be compactly supported. Nevertheless, for any \(a\geqslant 0\), by Theorem 2 and Corollary 1, we get \[\begin{align} \label{intRPhhh000} \int_{\mathbb{P}(\mathbb{V}) \times [0,a] } P h (x',t') \rho(dx',dt') & = \lim_{n\to\infty} \int_{\mathbb{R}_+} t \mathbb{E} \Big( P h (g_{n-1}\cdots g_{1} x, t + \sigma(g_{n-1}\cdots g_1, x)) ; \notag \\ & \qquad\qquad \tau_{x, t} >n-1, t+\sigma(g_{n-1}\cdots g_1, x))\leqslant a \Big) dt. \end{align}\tag{66}\] Now we notice that, since \(h\) has compact support and \(\mu\) has finite exponential moments, we may find constants \(\alpha, c >0\) such that for any \((x',t') \in \mathbb{P}(\mathbb{V}) \times \mathbb{R}\), \[\begin{align} | Ph(x',t') | \leqslant c e^{-\alpha t'} \end{align}\] so that, using the bound ?? , we arrive at \[\begin{align} \label{intRPhhh001} \int_{\mathbb{P}(\mathbb{V}) \times [a,\infty) } P h (x',t') \rho(dx',dt') \leqslant\int_{\mathbb{P}(\mathbb{V}) \times [a,\infty) } c (1+t') e^{-\alpha t'} dt' \leqslant c e^{-\alpha a}. \end{align}\tag{67}\] Besides, by Lemma 3 and 60 , we deduce that, for \(n \geqslant 1\), \[\begin{align} \label{intRPhhh002} & \int_{\mathbb{R}_+} t \mathbb{E} \left( P h (g_{n-1}\cdots g_{1} x, t + \sigma(g_{n-1}\cdots g_1, x)) ; \tau_{x,t} >n-1, t + \sigma(g_{n-1}\cdots g_1, x)) > a \right) dt \notag \\ & \leqslant c \int_{\mathbb{R}_+} t \mathbb{E} \left( e^{-\alpha (t + \sigma(g_{n-1}\cdots g_1, x) ) } ; \tau_{x,t} >n-1, t+\sigma(g_{n-1}\cdots g_1, x)) > a \right) dt \notag \\ & = c \int_a^\infty e^{-\alpha t} U_{n-1}^{x,n-1,\mathbf{1}} (t) dt \notag \\ & \leqslant c \int_a^\infty e^{-\alpha t} (1+t) dt \leqslant c e^{-\alpha a}, \end{align}\tag{68}\] where for the last line we used Corollary 8. Combining 66 , 67 and 68 yields \[\begin{align} & \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}_+ } P h (x',t') \rho(dx',dt') \notag \\ & = \lim_{n\to\infty} \int_{\mathbb{R}_+} t \mathbb{E} \Big( P h (g_{n-1}\cdots g_{1} x, t + \sigma(g_{n-1}\cdots g_1, x)); \tau_{x,t} >n-1 \Big) dt, \end{align}\] which together with 65 implies \(\rho(h)=\rho'(h)\). As the Radon measures \(\rho\) and \(\rho'\) are equal on continuous compactly supported functions, they are also equal on non-negative Borel measurable functions, which is the assertion of Corollary 2. ◻

Proof of Corollary 3. The assertions 12 and 13 follow from Proposition 8. To show 14 , we choose a non-negative continuous function \(\psi\) on \(\mathbb{R}\), supported in \([0,1]\), with \(\int_{0}^{1} \psi(t) dt =1\). Then, as \(W\) is non-decreasing, for any \(t > 0\), we deduce that \[\begin{align} \frac{W(t)}{t} \leqslant\frac{1}{t} \int_{\mathbb{R}} \psi(t'-t) W(t')dt' = \frac{1}{t} \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} \psi(t'-t) \rho(dx',dt') \leqslant\frac{W(t+1)}{t}. \end{align}\] Since, by 12 , we have \[\begin{align} \lim_{t\to\infty} \frac{1}{t} \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R}} \psi(t'-t) \rho(dx',dt') =\int_{0}^{1} \psi(t) dt =1, \end{align}\] the assertion 14 follows. ◻

Proof of Corollary 4. We shall first prove 15 . We claim that there exists \(c>0\) such that for any \(s \geqslant 0\), one has \[\begin{align} \label{rho-integrability-001} \rho(\mathbb{P}(\mathbb{V}) \times (-\infty, -s]) \leqslant c e^{-\alpha s}, \end{align}\tag{69}\] where \(\alpha>0\) is the exponent from 1 . Indeed, for \(0 \leqslant s < s'\), consider the function \[\begin{align} h_{s, s'}(x, t) = \mathbb{P} \left( t + \sigma(g, x) \in [-s', -s] \right) \mathbb{1}_{\{t \geqslant 0\}}. \end{align}\] Then, by 10 , we get that for any \(x \in \mathbb{P}(\mathbb{V})\), \[\begin{align} \rho(\mathbb{P}(\mathbb{V}) \times [-s', -s]) & = \lim_{n \to \infty} \int_{0}^{\infty} t \mathbb{P} \left( t + \sigma(g_n \cdots g_1, x) \in [-s', -s], \tau_{x, t} > n-1 \right) dt \notag\\ & = \lim_{n \to \infty} \int_{0}^{\infty} t \mathbb{E} \Big[ h_{s, s'} (g_{n-1} \cdots g_1 x, t + \sigma(g_{n-1} \cdots g_1, x) ); \tau_{x, t} > n-2 \Big] dt, \end{align}\] where in the last equality we used the Markov property. By our moment assumption 1 and Markov’s inequality, there exists \(c>0\) such that, for any \(x \in \mathbb{P}(\mathbb{V})\) and \(t \geqslant 0\), \[\begin{align} h_{s, s'}(x, t) \leqslant c e^{-\alpha (s+t)}. \end{align}\] Thus we get \[\begin{align} \rho(\mathbb{P}(\mathbb{V}) \times [-s', -s]) & \leqslant c e^{-\alpha s} \limsup_{n \to \infty} \int_{0}^{\infty} t \mathbb{E} \Big[ e^{-\alpha ( t + \sigma(g_{n-1} \cdots g_1, x) )}; \tau_{x, t} > n-2 \Big] dt \notag\\ & = c e^{-\alpha s} \limsup_{n \to \infty} \int_{0}^{\infty} e^{-\alpha t} U_{n-1}^{x, n-1, \mathbf{1}}(t) dt, \end{align}\] where the last equality holds due to Lemma 3. By Corollary 8 (see the proof of Proposition 9 for the reason why this result can be applied here), we get \[\begin{align} \rho(\mathbb{P}(\mathbb{V}) \times [-s', -s]) \leqslant c e^{-\alpha s} \int_{0}^{\infty} e^{-\alpha t} (1 + t) dt \leqslant c' e^{-\alpha s}. \end{align}\] By letting \(s' \to \infty\), we obtain 69 . Since the function \(W\) is non-decreasing, for any \(s\geqslant 0\), we have \[\begin{align} W(-s-1) \leqslant\rho(\mathbb{P}(\mathbb{V}) \times [-s-1, -s]) \leqslant c' e^{-\alpha s} \end{align}\] and 15 follows.

To conclude the proof of the corollary, it remains to show that \(\rho( \mathbb{P}(\mathbb{V}) \times (-\infty,0) ) >0\). For \(x\in \mathbb{P}(\mathbb{V})\) and \(t\in \mathbb{R}\), set \(h_0(x,t) = \mathbb{1}_{\{t < 0\}}\) and, for \(n\geqslant 1\), define \(h_n(x,t) = \mathbb{P} \left(\tau_{x,t} = n \right)\). By Corollary 2, for any \(n\geqslant 1\), we have \[\begin{align} \int_{\mathbb{P}(\mathbb{V}) \times \mathbb{R} } h_{n-1} (x,t) \rho(dx,dt) = \int_{\mathbb{P}(\mathbb{V}) \times [0, \infty) } h_{n} (x,t) \rho(dx,dt). \end{align}\] This gives \[\begin{align} \rho( \mathbb{P}(\mathbb{V}) \times (-\infty,0) ) = \int_{\mathbb{P}(\mathbb{V}) \times [0, \infty) } \mathbb{P} \left(\tau_{x,t} = n \right) \rho(dx,dt). \end{align}\] Since, for any \(x\in \mathbb{P}(\mathbb{V})\) and \(t\in \mathbb{R}\), the stopping time \(\tau_{x,t}\) is \(\mathbb{P}\)-almost surely finite, it follows that \[\begin{align} \mathbb{P}(\mathbb{V}) \times [0,\infty) = \bigcup_{n\geqslant 1} \{ (x,t) \in \mathbb{P}(\mathbb{V}) \times [0,\infty) : \mathbb{P} \left(\tau_{x,t} = n \right) >0 \}. \end{align}\] By 14 of Corollary 3, we know that \(\rho\) is not zero on \(\mathbb{P}(\mathbb{V}) \times [0,\infty)\). Therefore, there exists \(n\geqslant 1\) such that \[\begin{align} \rho\left( \{ (x, t) \in \mathbb{P}(\mathbb{V}) \times [0,\infty) : \mathbb{P} \left(\tau_{x, t} = n \right) >0 \} \right) > 0, \end{align}\] which implies that \[\begin{align} \int_{\mathbb{P}(\mathbb{V}) \times [0,\infty) } \mathbb{P} \left(\tau_{x, t} = n \right) \rho(dx, dt) > 0. \end{align}\] The conclusion follows. ◻

Proof of Corollary 8. Applying Theorem 10 with \(\theta = 1\), we get that for any \(n \geqslant 1\) and \(t \in \mathbb{R}\), \[\begin{align} U^{\mathfrak f}_n(t) \leqslant U^{\mathfrak f}_{1}(t + A) + A \left( 1 + \max \{t,0\} \right). \end{align}\] From our moment assumptions, we have \(U^{\mathfrak f}_{1}(t + A) \leqslant c \left( 1 + \max \{t,0\} \right)\) and the conclusion follows. ◻

Proof of Corollary 7. We fix a non-negative continuous compactly supported function \(\varphi\) on \(\mathbb{R}\). Let \(b, \gamma, A > 0\) be as in Theorem 10. Then, for any \(1\leqslant n\leqslant m\), we have \[\begin{align} & \int_{\mathbb{R}} U_n^{\mathfrak f, \theta} (t) \varphi(t + A n^{-\gamma}) dt \notag\\ &\leqslant\int_{\mathbb{R}} U_m^{\mathfrak f, \theta} \left(t+ A n^{-\gamma}\right) \varphi(t + A n^{-\gamma}) dt + A n^{-b} \int_{\mathbb{R}} \left( 1 + \max \{t,0\} \right) \varphi(t + A n^{-\gamma}) dt \notag\\ &= \int_{\mathbb{R}} U_m^{\mathfrak f, \theta} (t) \varphi(t) dt + A n^{-b} \int_{\mathbb{R}} \left( 1 + \max \{t,0\} \right) \varphi(t + A n^{-\gamma}) dt. \end{align}\] Taking the limit as \(m \to\infty\), we obtain that, for any \(n\geqslant 1\), \[\begin{align} \label{eq32sup-inf-001} & \int_{\mathbb{R}} U_n^{\mathfrak f, \theta} (t) \varphi(t + A n^{-\gamma}) dt \notag\\ & \leqslant\liminf_{m\to\infty} \int_{\mathbb{R}} U_m^{\mathfrak f, \theta} (t) \varphi(t) dt + A n^{-b} \int_{\mathbb{R}} \left( 1 + \max \{t,0\} \right) \varphi(t + A n^{-\gamma}) dt. \end{align}\tag{90}\] By the continuity of \(\varphi\) and the uniform bound of Corollary 8, we have that, as \(n\to\infty\), \[\begin{align} \int_{\mathbb{R}} U_n^{\mathfrak f, \theta} (t) \varphi(t + A n^{-\gamma}) dt - \int_{\mathbb{R}} U_n^{\mathfrak f, \theta} (t) \varphi(t) dt \to 0. \end{align}\] Hence, taking the limit as \(n\to\infty\) in 90 , we get \[\begin{align} \label{eq32sup-inf-002} \limsup_{n\to\infty} \int_{\mathbb{R}} U_n^{\mathfrak f, \theta} (t) \varphi(t) dt \leqslant\liminf_{m\to\infty} \int_{\mathbb{R}} U_m^{\mathfrak f, \theta} (t) \varphi(t) dt. \end{align}\tag{91}\] Therefore, the integral \(\int_{\mathbb{R}} U_n^{\mathfrak f, \theta} (t) \varphi(t) dt\) admits a limit as \(n\to\infty\), and this limit is finite by Corollary 8. Hence, the assertion 87 follows from Lemma 5.

Now we prove the assertion 88 . First, note that, for any \(n \geqslant 1\), we have \[\begin{align} \label{convergence-U-n-t-001} \lim_{t \to \infty} \frac{1}{t} U_n^{\mathfrak f, \theta}(t) = \mathbb{E} \theta. \end{align}\tag{92}\] Indeed, by 82 , we write, for \(t \geqslant 0\), \[\begin{align} \big| U_n^{\mathfrak f, \theta}(t) - t \mathbb{E} \theta \big| \leqslant t \|\theta\|_{\infty} \int_{\mathbb{X}} \mathbb{P} \big( \tau_{x,t}^{\mathfrak f} \leqslant n \big) \nu(dx) + \|\theta\|_{\infty} \int_{\mathbb{X}} \mathbb{E} \big| \tilde{S}^{x}_n \big| \nu(dx). \end{align}\] By 80 , we have \[\begin{align} \int_{\mathbb{X}} \mathbb{P} \big( \tau_{x,t}^{\mathfrak f} \leqslant n \big) \nu(dx) \leqslant\int_{\mathbb{X}} \mathbb{P} \Big( \max_{1\leqslant k\leqslant n}|\tilde{S}^{x}_k| >t \Big) \nu(dx) \leqslant\sum_{k=1}^n \int_{\mathbb{X}} \mathbb{P} \big( \big| \tilde{S}^{x}_k \big| > t \big) \nu(dx), \end{align}\] which implies that \[\begin{align} \lim_{t \to \infty} \int_{\mathbb{X}} \mathbb{P} \big( \tau_{x,t}^{\mathfrak f} \leqslant n \big) \nu(dx) = 0. \end{align}\] Hence 92 holds. To prove 88 , we fix a continuous compactly supported function \(\varphi\) on \(\mathbb{R}\) which is non-negative with support in \([0, 1]\). We also assume that \(\int_0^1 \varphi(t) dt = 1\). In particular, for any \(t \in \mathbb{R}\), we have \[\begin{align} U^{\mathfrak f, \theta}(t) = \int_{\mathbb{R}} \varphi(t'-t) U^{\mathfrak f, \theta}(t) dt' \leqslant\int_{\mathbb{R}} \varphi(t'-t) U^{\mathfrak f, \theta}(t') dt'. \end{align}\] By Theorem 10, for any \(n \geqslant 1\) and \(t \geqslant 0\), we derive that \[\begin{align} \int_{\mathbb{R}} \varphi(t'-t) U^{\mathfrak f, \theta}(t') dt' & = \lim_{m \to \infty} \int_{\mathbb{R}} \varphi(t'-t) U_m^{\mathfrak f, \theta}(t') dt' \notag\\ & \leqslant\int_{\mathbb{R}} \varphi(t'-t) U_{n}^{\mathfrak f, \theta}(t' + A n^{-\gamma}) dt' + A n^{-b} \int_{\mathbb{R}} \varphi(t'-t) (1 + t') dt' \notag\\ & \leqslant\sup_{t \leqslant t' \leqslant t + 1 + A n^{-\gamma}} U_{n}^{\mathfrak f, \theta}(t') + A (t +2) n^{-b}. \end{align}\] From 92 , we get \[\begin{align} \limsup_{t \to \infty} \frac{1}{t} U^{\mathfrak f, \theta}(t) \leqslant\mathbb{E} \theta + A n^{-b}. \end{align}\] As this is true for any \(n \geqslant 1\), we get \(\limsup_{t \to \infty} \frac{1}{t} U^{\mathfrak f, \theta}(t) \leqslant\mathbb{E} \theta\). Reasoning in the same way, we can prove that \(\liminf_{t \to \infty} \frac{1}{t} U^{\mathfrak f, \theta}(t) \geqslant\mathbb{E} \theta\) and so 88 holds. ◻

Proof. Since the sequence \((S^x_n)_{n \geqslant 0}\) is a zero-mean martingale, we can apply the optional stopping theorem to obtain the following expression for \(V_n(x, t)\): \[\begin{align} \label{optional-stp-iden} V_n(x, t) = \mathbb{E} \left( t + S_n^x; \vartheta_{x,t} > n \right) &= t - \mathbb{E} \left( t + S_n^x; \vartheta_{x,t} \leqslant n \right) \notag \\ &= t - \mathbb{E} \left( t + S_{ \vartheta_{x,t} }^x; \vartheta_{x,t} \leqslant n \right). \end{align}\tag{95}\] The lower bound follows as the random variable \(t + S_{ \vartheta_{x,t} }^x\) is negative. The upper bound also follows from 95 since \(- \mathbb{E} ( t + S_{ \vartheta_{x,t} }^x; \vartheta_{x,t} \leqslant n )\) is increasing in \(n\). ◻

Proof. By Proposition 12, for any \(n \geqslant 1\), \(x\in \mathbb{X}\) and \(t \in \mathbb{R},\) we get \[\begin{align} V_{n} (x, t) \leqslant V_{1} (x, t) + A (1+ \max\{t, 0\}). \end{align}\] Recall that \(V_1(x, t) = \mathbb{E} \left( t + \sigma(g_1, x); t + \sigma(g_1, x) \geqslant 0 \right).\) If \(t \leqslant 0\), we have \(V_{1} (x, t) \leqslant\mathbb{E} |\sigma(g_1, x)| \leqslant c\). If \(t> 0\), we have \(V_{1} (x, t) \leqslant t + \mathbb{E} |\sigma(g_1, x)| \leqslant t + c\). The conclusion follows. ◻

Proof. Using the Markov property and the definition of the function \(V_n\) in 94 , we get \[\begin{align} V_n(x, t) = \mathbb{E} \Big( V_{n-1} (g_1 x, t + \sigma(g_1, x)); t + \sigma(g_1, x) \geqslant 0 \Big), \end{align}\] where for \(n = 1\) we have written \(V_0(x, t) = \max\{t, 0\}\). By Corollary 9, there exists a constant \(c > 0\) such that for any \(n \geqslant 1\), \(x\in \mathbb{X}\) and \(t \leqslant 0,\) \[\begin{align} V_n(x, t) \leqslant c \mathbb{E} \Big( t + \sigma(g_1, x); t + \sigma(g_1, x) \geqslant 0 \Big) & = c \int_{0}^{\infty} \mathbb{P} \left( t + \sigma(g_1, x) \geqslant s \right) ds \notag\\ & = c \int_{-t}^{\infty} \mathbb{P} \left( \sigma(g_1, x) \geqslant s \right) ds. \end{align}\] The conclusion now follows from the assumption 71 and Markov’s inequality. ◻

Proof. Let \(m = [\gamma^{-2} M^2]\) and \(K= \left[ n/ m \right]\), where \(\gamma > 0\) will be chosen later. It is easy to see that, for any \(x\in \mathbb{X}\) and \(t \in \mathbb{R}\), \[\begin{align} \mathbb{P}\left( \max_{1\leqslant k\leqslant n}\left\vert t + S^x_{k}\right\vert \leqslant M \right) \leqslant\mathbb{P}\left( \max_{1\leqslant k\leqslant K}\left\vert t+S^x_{km}\right\vert \leqslant M \right) . \label{nu000} \end{align}\tag{97}\] Using the cocycle property 70 , it follows that \[\begin{align} \mathbb{P}\left( \max_{1\leqslant k\leqslant K}\left\vert t +S^x_{km}\right\vert \leqslant M \right) \leqslant\mathbb{P}\left( \max_{1\leqslant k\leqslant K-1}\left\vert t +S^x_{km}\right\vert \leqslant M \right) \sup_{x'\in \mathbb{X}} \sup_{t'\in \mathbb{R}} \mathbb{P} \left( \left\vert t'+S^{x'}_{m}\right\vert \leqslant M \right) , \end{align}\] from which iterating, we get \[\begin{align} \label{Piterations-001} \mathbb{P} \left( \max_{1\leqslant k\leqslant K}\left\vert t +S^x_{km}\right\vert \leqslant M \right) \leqslant\left( \sup_{x'\in \mathbb{X}} \sup_{t'\in \mathbb{R}} \mathbb{P} \left( \left\vert t'+S^{x'}_{m}\right\vert \leqslant M \right) \right) ^{K}. \end{align}\tag{98}\] By the effective central limit theorem 79 , there exist constants \(c, \mathbf{v}, \epsilon > 0\) such that for any \(x' \in \mathbb{X}\), \(t' \in \mathbb{R}\) and \(m \geqslant 1\), \[\begin{align} \mathbb{P}\left( \left| t' +S^{x'}_{m} \right| \leqslant M \right) = \mathbb{P} \left( \frac{S^{x'}_{m}}{\sqrt{m}} \in \left[ \frac{-M - t' }{\sqrt{m}}, \frac{M- t'}{\sqrt{m}} \right] \right) \leqslant\int_{ \frac{-M-t'}{\sqrt{m}} }^{ \frac{M-t'}{\sqrt{m} } } \phi_{\mathbf{v}^2} (u) du + \frac{c}{ m^{\epsilon} } \leqslant\frac{1}{2}, \end{align}\] where in the last inequality we take \(\gamma >0\) sufficiently small so that \(\frac{2M}{\sqrt{m}} + \frac{c}{ m^{\epsilon} } \leqslant 4 \gamma + c (2 \gamma^2)^{\epsilon} \leqslant\frac{1}{2}\). The assertion of the lemma follows from 98 . ◻

Proof of Lemma 7. By Lemma 8, there exists a constant \(\beta>0\) such that for any \(n \geqslant 1\), \(\ell \geqslant 1\), \(x \in \mathbb{X}\) and \(t \in \mathbb{R}\), \[\begin{align} \mathbb{P} \left( \nu_{n,x,t}> \ell \right) = \mathbb{P} \left( \sup_{ 1\leqslant k \leqslant\ell } |t + S_k^{x} | \leqslant 2 n^{1/2-\varepsilon} \right) \leqslant 2 \exp \left(- \frac{\beta \ell}{4 n^{1-2\varepsilon}}\right), \end{align}\] completing the proof of the lemma, by replacing \(\beta/4\) with \(\beta\). ◻

Proof. Set \(A_{n,x,t} = \{ \max_{ k \in [1, n^{1/2-\varepsilon} - t] } |\sigma(g_{k},g_{k-1} \cdots g_{1} x)| \leqslant n^{\varepsilon/2} \}\). By 71 , there exist constants \(c, c_{\varepsilon}, \alpha > 0\) such that for any \(n\geqslant 1\), \(x\in \mathbb{X}\) and \(t\geqslant- n^{\varepsilon}\), \[\begin{align} \mathbb{P} \big( A_{n,x,t}^c \big) \leqslant c \max\{ n^{1/2-\varepsilon} - t, 0 \} e^{- \alpha n^{\varepsilon/2}} \leqslant c e^{- c_{\varepsilon} n^{\varepsilon/2}}. \end{align}\] On the event \(A_{n,x,t}\), it holds that for any \(k \in [1, n^{1/2-\varepsilon} - t]\), \[\begin{align} |t + S_k^x| \leqslant|t| + k n^{\varepsilon/2} \leqslant n^{1/2 - \varepsilon/2}. \end{align}\] Hence, by the definition of \(\nu_{n,x,t}\), it follows that \(\{ \nu_{n,x,t} \leqslant n^{1/2-\varepsilon} - t \}\) is included in \(A_{n,x,t}^c\). The lemma follows. ◻

Proof. Consider the event \[\begin{align} A_{n,x} = \left\{ \max_{1 \leqslant k \leqslant n}\left\vert \sigma(g_k, g_{k-1} \cdots g_1 x) \right\vert \leqslant n^{1/2-2\varepsilon} \right\}. \end{align}\] We have \[\begin{align} \mathbb{E} \left( \left\vert t + S_{ \vartheta_{x, t} }^x \right\vert; \vartheta_{x, t} \leqslant n \right) = \mathbb{E} \left( \left\vert t + S_{ \vartheta_{x, t} }^x \right\vert; \vartheta_{x, t} \leqslant n, A_{n,x} \right) + \mathbb{E} \left( \left\vert t + S_{ \vartheta_{x, t} }^x \right\vert; \vartheta_{x, t} \leqslant n, A_{n,x}^c \right). \label{eq-lemma1-000-0} \end{align}\tag{99}\] For the first term, since \(\vartheta_{x, t}\) is the first integer \(k \geqslant 1\) when \(t + \sum_{i=1}^{k} \sigma(g_k, g_{k-1} \cdots g_1 x)\) becomes negative and the size of the jump is bounded by \(|\sigma(g_{\vartheta_{x, t}}, g_{ \vartheta_{x, t} -1} \cdots g_1 x)|\) which does not exceed \(n^{1/2-2\varepsilon}\) on the event \(A_{n,x}\), we get that for \(t \geqslant n^{1/2-\varepsilon}\), \[\begin{align} \label{eq-lemma1-R1-0} \mathbb{E} \left( \left\vert t + S_{ \vartheta_{x, t} }^x \right\vert; \vartheta_{x, t} \leqslant n, A_{n,x} \right) \leqslant n^{1/2-2\varepsilon}\mathbb{P} \left( A_{n,x}\right) \leqslant n^{1/2-2\varepsilon} \leqslant{\frac{t}{n^{\varepsilon}}}. \end{align}\tag{100}\] To handle the second term in 99 , by Markov’s inequality and 71 , we get \[\begin{align} \mathbb{P} \left( A_{n,x}^c \right) &=\mathbb{P} \left( \max_{1 \leqslant k\leqslant n} |\sigma(g_k, g_{k-1} \cdots g_1 x)| > n^{1/2 - 2\varepsilon} \right) \notag \\ &\leqslant n \sup_{x\in \mathbb{X}} \mathbb{P} \left( |\sigma(g_{1}, x)| > n^{1/2-2\varepsilon} \right) \leqslant c n e^{- \alpha n^{1/2-2\varepsilon} } \leqslant c e^{- \alpha n^{1/3}}. \end{align}\] Therefore, by Hölder’s inequality, \[\begin{align} \label{eq-lemma1-R2-0} \mathbb{E} \left( \left\vert t + S_{ \vartheta_{x, t} }^x \right\vert; \vartheta_{x, t} \leqslant n, A_{n,x}^c \right) & = \sum_{k =1}^n \mathbb{E} \left( \left\vert t + S_{k}^x \right\vert; \vartheta_{x, t} = k, A_{n,x}^c \right) \notag\\ & \leqslant\sum_{k =1}^n \mathbb{E} \left( \left\vert t + S_{k}^x \right\vert; A_{n,x}^c \right) \notag\\ & \leqslant\sum_{k = 1}^n \mathbb{E}^{1/2} \left( \left\vert t + S_{k}^x \right\vert^2 \right) \mathbb{P}^{1/2}(A_{n,x}^c) \notag\\ & \leqslant c n (t + n^{1/2}) e^{- \frac{\alpha}{2} n^{1/3 } }, \end{align}\tag{101}\] where in the last inequality we used \[\begin{align} \mathbb{E}^{1/2} \left( \left\vert t + S_{k}^x \right\vert^2 \right) \leqslant t+ \sup_{x\in \mathbb{X}} \sqrt{\mathbb{E} (S_{k}^{x} )^{ 2 } } \leqslant t+c n^{1/2}, \end{align}\] by Minkowski’s inequality. Combining 100 and 101 completes the proof. ◻

Proof of Proposition 13. The main idea of the proof is to stop the process \((t+S_n^x)_{n\geqslant 1}\) at the exit time \(\nu_{n,x,t}\) and to apply the Markov property.

We shall first show that there exist constants \(c, \varepsilon_{0}>0\) such that for any \(\varepsilon\in (0, \varepsilon_0)\), \(n\geqslant 1\), \(x \in \mathbb{X}\) and \(t\geqslant n^{1/2-\varepsilon}\), \[\begin{align} \label{Un-xy-Bound-001-0} V_n(x, t) \leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) t. \end{align}\tag{102}\] Let \(x \in \mathbb{X}\) and \(t\geqslant n^{1/2-\varepsilon}\). Using 95 , we get \[\begin{align} V_n(x, t) = t - \mathbb{E} \left( t + S_{ \vartheta_{x, t} }^x; \; \vartheta_{x, t} \leqslant n\right) \leqslant t + \mathbb{E} \left( \left\vert t + S_{ \vartheta_{x, t} }^x \right\vert; \vartheta_{x, t} \leqslant n \right). \end{align}\] This inequality, combined with Lemma 11, yields the desired bound 102 .

Now we prove ?? by using the bound 102 and the Markov property. Note that for any \(\varepsilon>0\), \(n \geqslant 1\), \(x \in \mathbb{X}\) and \(t \in \mathbb{R}\), \[\begin{align} V_{n} (x, t) = \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} > n^{1-\varepsilon} \right) + \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} \leqslant n^{1-\varepsilon} \right). \label{bound32J143J2-0} \end{align}\tag{103}\] By the Cauchy-Schwarz inequality and Lemmas 7 and 10, we get \[\begin{align} \label{first-term-V-n-001} \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} > n^{1-\varepsilon} \right) \leqslant c' e^{- c n^{\varepsilon}} \left( 1 + \max\{t, 0\} \right). \end{align}\tag{104}\] For the second term in 103 , we decompose it as \[\begin{align} \label{bound32J132002-0} \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} \leqslant n^{1-\varepsilon} \right) = \sum_{k=1}^{ [n^{1-\varepsilon}] } \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} = k \right). \end{align}\tag{105}\] By the Markov property, we get \[\begin{align} \label{bound32E-0} \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} = k \right) = \mathbb{E} \left[ V_{n-k} \left( g_k \cdots g_1 x, t + S_k^x \right); \vartheta_{x, t} >k, \nu_{n,x, t} = k \right]. \end{align}\tag{106}\] On the event \(\{ \vartheta_{x, t} >k, \nu_{n,x, t} = k \}\), we have \(t + S_k^x \geqslant n^{1/2 - \varepsilon}\). Thus, by 102 , we get that for any \(k \in [1, [n^{1-\varepsilon}]]\), \[\begin{align} \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} = k \right) \leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) \mathbb{E} \left( t + S_k^x; \vartheta_{x, t} >k, \nu_{n,x, t} = k \right). \end{align}\] Inserting this into 105 , we obtain \[\begin{align} \label{equ-n-k-S} \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} \leqslant n^{1-\varepsilon} \right) \leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) \sum_{k=1}^{ [n^{1-\varepsilon}] } \mathbb{E} \left( t + S_k^x; \vartheta_{x, t} >k, \nu_{n,x, t} = k \right). \end{align}\tag{107}\]

We claim that the sequence \(\big( (t + S_k^x) \mathbb{1}_{\{ \vartheta_{x, t} \leqslant k \}} \big)_{k \geqslant 1}\) is a supermartingale with respect to the natural filtration \((\mathscr A_k)_{k \geqslant 1}\). Indeed, for any \(k \geqslant 1\) and \(A \in \mathscr A_k\), we have \[\begin{align} \mathbb{E} \left( t + S_{k+1}^x; \vartheta_{x, t} \leqslant k+1, A \right) & = \mathbb{E} \left( t + S_{k+1}^x; \vartheta_{x, t} \leqslant k, A \right) + \mathbb{E} \left( t + S_{k+1}^x; \vartheta_{x, t} = k+1, A \right) \notag\\ & \leqslant\mathbb{E} \left( t + S_{k}^x; \vartheta_{x, t} \leqslant k, A \right), \end{align}\] where we have used 72 and the fact that, by the definition of \(\vartheta_{x, t}\), one has \(t + S_{k+1}^x < 0\) on the set \(\{ \vartheta_{x, t} = k+1 \}\). Still by 72 , this implies that the sequence \(\big( (t + S_k^x) \mathbb{1}_{\{ \vartheta_{x, t} > k \}} \big)_{k \geqslant 1}\) is a submartingale. From this submartingale property, we have that, for any \(k\in [1, [n^{1-\varepsilon}] ]\), \[\begin{align} \label{Equ-desired-0} \mathbb{E} \left( t + S_k^x; \vartheta_{x, t} >k, \nu_{n,x, t} = k \right) \leqslant\mathbb{E} \left( t + S_{[n^{1-\varepsilon}]}^x; \vartheta_{x, t} > [n^{1-\varepsilon}], \nu_{n,x, t} = k \right). \end{align}\tag{108}\] Therefore, using 107 and 108 , we get \[\begin{align} \mathbb{E} \left( t + S_{n}^x; \vartheta_{x, t} > n, \nu_{n,x, t} \leqslant n^{1-\varepsilon} \right) & \leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) \mathbb{E} \left( t + S_{[n^{1-\varepsilon}]}^x; \vartheta_{x, t} > [n^{1-\varepsilon}], \nu_{n,x, t} \leqslant n^{1-\varepsilon} \right) \notag\\ & \leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) V_{[n^{1-\varepsilon}]}(x, t). \end{align}\] Combining this with 103 and 104 completes the proof of Proposition 13. ◻

Proof. Define the sequence \((m_j)_{j\geqslant 0}\) as follows: with \(m_0=m\), and for \(j \geqslant 0\), \[\begin{align} \label{def-mj-sequence} m_{j+1} = [m_j^{1-\varepsilon}]. \end{align}\tag{112}\] This sequence is non-increasing and converges to \(1\). Fix \(2 \leqslant n < m\). There exists a least integer \(\ell \in \mathbb{N}\) such that \(m_{\ell} \leqslant n\) and we have \(\ell \leqslant\alpha \log \frac{\log m}{\log n}\) for some constant \(\alpha >0\). By our assumption 111 , it holds that for any \(j \in \mathbb{N}\) and \(t \in \mathbb{R}\), \[\begin{align} V_{m_j} (t) \leqslant\left( 1 + b m_j^{-\varepsilon} \right) V_{m_{j+1} } \left( t + a m_j^{-\varepsilon} \right) + b m_j^{-\beta} (1 + \max \{t, 0 \}). \end{align}\] By iterating over \(j\), we get \[\begin{align} V_{m_j} (t) & \leqslant\left( 1 + b m_{j+1}^{-\varepsilon} \right) \left( 1 + b m_j^{- \varepsilon} \right) V_{m_{j+2} } \left( t + a m_{j}^{-\varepsilon} + a m_{j+1}^{-\varepsilon} \right) \notag\\ & \quad + b m_{j+1}^{- \beta} \left( 1 + b m_j^{-\varepsilon} \right) \left(1+ \max \{t,0\} + a m_{j}^{-\varepsilon} \right) + b m_j^{-\beta} (1+ \max \{t,0\} ). \end{align}\] After \(\ell - 1\) iterations, we obtain \[\begin{align} \label{V-A-B-j} V_{m} (t) \leqslant A_{\ell} V_{m_{\ell} } \left(t + a B_{\ell} \right) + b \sum_{j=0}^{\ell -1} m_j^{- \beta} A_{j} \left(1+\max \{t,0\} + a B_j \right), \end{align}\tag{113}\] where, for \(0 \leqslant j \leqslant\ell\), \[\begin{align} A_j = \prod_{k=0}^{j-1} \left( 1 + b m_{k}^{-\varepsilon} \right), \qquad B_j = \sum_{k=0}^{j-1} m_k^{-\varepsilon} \end{align}\] with the convention \(\prod_{k=0}^{-1} = 1\) and \(\sum_{k=0}^{-1} = 0\). To give an upper bound for \(A_{\ell}\), we first use the inequality \(1 + x \leqslant e^x\) for \(x \geqslant 0\) to get \[\begin{align} \label{majAm} A_{\ell} \leqslant\prod_{k=0}^{\ell -1} e^{b m_k^{-\varepsilon} } = e^{b B_{\ell}}. \end{align}\tag{114}\] Since \(\ell \in \mathbb{N}\) is the least integer such that \(m_{\ell} \leqslant n\), we have \(m_{\ell - 1} \geqslant n\), so that \[\begin{align} \label{bound-B-ell-sum} B_{\ell} = \sum_{k=0}^{\ell-1} m_k^{-\varepsilon} & = \frac{1}{m^{\varepsilon}} + \frac{1}{m_{1}^{\varepsilon}} + \ldots + \frac{1}{m_{\ell-1}^{\varepsilon}} \notag\\ & \leqslant\frac{c}{n^{\varepsilon}} m^{\varepsilon(1-\varepsilon)^{\ell -1} } \left( \frac{1}{m^{\varepsilon}} + \frac{1}{m^{\varepsilon(1-\varepsilon) }} + \ldots + \frac{1}{m^{\varepsilon(1-\varepsilon)^{\ell -1} }} \right) \notag\\ & \leqslant\frac{c}{n^{\varepsilon}} \left( m^{\varepsilon((1-\varepsilon)^{\ell -1} -1 )} + m^{\varepsilon(1-\varepsilon) ((1-\varepsilon)^{\ell -2} -1 )} + \ldots + 1 \right) \notag\\ & = \frac{c}{n^{\varepsilon}} \sum_{k=0}^{\ell -1} m^{\varepsilon(1-\varepsilon)^{k} ((1-\varepsilon)^{\ell -1- k} -1 ) } \notag\\ & = \frac{c}{n^{\varepsilon}} \sum_{k=0}^{\ell -1} m^{\varepsilon(1-\varepsilon)^{\ell - 1-k} ((1-\varepsilon)^{k} -1 ) }. \end{align}\tag{115}\] As \(\min_{ 0 \leqslant k \leqslant\ell -2 } m_k \geqslant n\), it holds that \(\min_{ 0 \leqslant k \leqslant\ell -2 } m^{(1-\varepsilon)^{\ell - 2 -k}} \geqslant n\) and for any \(0 \leqslant k \leqslant\ell -2\), \[\begin{align} \frac{m^{\varepsilon(1-\varepsilon)^{\ell - 2 -k} ((1-\varepsilon)^{k+1} -1 ) } }{ m^{\varepsilon(1-\varepsilon)^{\ell - 1-k} ((1-\varepsilon)^{k} -1 ) } } = m^{ - \varepsilon^2 (1 - \varepsilon)^{\ell -2 -k} } \leqslant n^{- \varepsilon^2 }. \end{align}\] Hence, we obtain \[\begin{align} \sum_{k=0}^{\ell -1} m^{\varepsilon(1-\varepsilon)^{\ell - 1-k} ((1-\varepsilon)^{k} -1 ) } \leqslant\sum_{k=0}^{\ell -1} n^{- \varepsilon^2 k} \leqslant\frac{1}{1 - n^{-\varepsilon^2}} \leqslant\frac{1}{1 - 2^{-\varepsilon^2}}. \end{align}\] Substituting this into 115 and 114 , we get \[\begin{align} \label{bound-A-B-ell-24} B_{\ell} \leqslant\frac{c}{n^{\varepsilon}}, \quad A_{\ell} \leqslant e^{b B_{\ell}} \leqslant 1 + \frac{c'}{n^{\varepsilon}}, \end{align}\tag{116}\] where \(c' > 0\) depends on \(b\). Reasoning as above shows that \(\sum_{j=0}^{\ell -1} m_j^{-\beta} \leqslant c n^{-\beta}\). Therefore, from 113 and 116 , we obtain that there exists a constant \(c > 0\) depending on \(a\) and \(b\) such that \[\begin{align} \label{Bound-Vmt-aaa} V_{m} (t) & \leqslant\left( 1 + c n^{-\varepsilon} \right) V_{m_{\ell} } \left( t + c n^{-\varepsilon} \right) + c \left(1+\max \{t,0\} \right) \sum_{j=0}^{\ell -1} m_j^{-\beta} \notag\\ & \leqslant\left( 1 + c n^{-\varepsilon} \right) V_{m_{\ell} } \left( t + c n^{-\varepsilon} \right) + c n^{- \beta} \left(1+\max \{t,0\} \right). \end{align}\tag{117}\] As \(m_{\ell} \leqslant n\), by 109 , we also have \[\begin{align} \label{Bound-Vmt-bbb} V_{m_{\ell} } \left( t + c n^{-\varepsilon} \right) & \leqslant V_{n} \left( t + c n^{-\varepsilon} + a m_{\ell}^{-\varepsilon} \right) + c m_{\ell}^{-\beta} \left( 1 + \max \{t, 0\} \right) \notag\\ & \leqslant V_{n} \left( t + c n^{-\varepsilon(1-\varepsilon)} \right) + c n^{-\beta(1-\varepsilon)} \left( 1 + \max \{t,0\} \right), \end{align}\tag{118}\] where in the last inequality we used the fact that \(n \leqslant m_{\ell -1}\), so that \(n^{1-\varepsilon} \leqslant m_{\ell -1}^{1 - \varepsilon} \leqslant m_{\ell} + 1\) by 112 . Substituting 118 into 117 gives that for any \(1 \leqslant n \leqslant m\) and \(t \in \mathbb{R}\), \[\begin{align} \label{inequ-recursive-001} V_{m} (t) \leqslant\left( 1 + c n^{-\varepsilon} \right) V_{n} \left( t + c n^{-\varepsilon(1-\varepsilon)} \right) + c n^{- \beta (1-\varepsilon) } (1+\max \{t,0\}). \end{align}\tag{119}\] In particular, using the fact that \(V_1(t) \leqslant b (1 + \max\{t, 0\})\), we get \[\begin{align} V_{n} (t) \leqslant\left( 1 + c \right) V_{1} \left( t + c \right) + c (1+\max \{t,0\}) \leqslant c' (1+\max \{t,0\}). \end{align}\] Finally, inserting this into 119 yields the conclusion of the lemma. The proof of the fact that \(A=0\) when \(a=0\) is left to the reader. ◻

Proof of Proposition 12. It follows from Lemma 6 and Proposition 13 that the assumptions of Lemma 12 are satisfied. Hence the conclusion follows. ◻

Proof of Proposition 11. Since \(\mathbb{P} \left( \vartheta_{x,t} > n \right)\) is non-decreasing in \(t \in \mathbb{R}\), it suffices to prove the result for \(t \geqslant 0\). By Corollary 9, we can find a constant \(c>0\) such that for any \(n \geqslant 1\), \(x \in \mathbb{X}\) and \(t \in \mathbb{R}\), \[\begin{align} V_n(x,t) = \mathbb{E} \left( t + S_n^x; \vartheta_{x,t} > n \right) \leqslant c (1 + \max\{t, 0\}). \end{align}\] We fix \(\beta >0\), whose value will be determined later. Note that we can assume \(t \leqslant n^{\beta}\). We write \[\begin{align} \mathbb{P} \left( \vartheta_{x,t} > n \right) = \mathbb{P} \left( \vartheta_{x,t} > n, |S_n^x| \leqslant n^{2\beta} \right) + \mathbb{P} \left( \vartheta_{x,t} > n, |S_n^x| > n^{2\beta} \right). \end{align}\] Recall that \(0 \leqslant t \leqslant n^{\beta}\). Then, on the set \(\{ \vartheta_{x,t} > n, |S_n^x| > n^{2\beta} \}\) we have \(t + S_n^x \geqslant n^{2\beta}\). From the above, we get that, by Chebyshev’s inequality, \[\begin{align} \mathbb{P} \left( \vartheta_{x,t} > n \right) \leqslant\mathbb{P} \left( |S_n^x| \leqslant n^{2\beta} \right) + n^{-2 \beta} V_n(x, t) \leqslant\mathbb{P} \left( |S_n^x| \leqslant n^{2\beta} \right) + c n^{- 2\beta} (1 + \max\{t, 0\}). \end{align}\] Now we apply the effective central limit theorem 79 to obtain \[\begin{align} \mathbb{P} \left( |S_n^x| \leqslant n^{2\beta} \right) \leqslant\int_{-n^{2\beta - \frac{1}{2}} }^{ n^{2\beta - \frac{1}{2}} } \phi_{\mathbf{v}^{2}}(u) du + \frac{c}{n^{\epsilon}} \leqslant c \left( n^{2\beta - \frac{1}{2}} + n^{-\epsilon} \right), \end{align}\] which leads to the desired result as soon as \(\beta < \frac{1}{4}\). ◻

Proof. For brevity, we denote for \(n > k \geqslant 1\) and \(t \in \mathbb{R}\), \[\begin{align} A_{k,n}(t) = \bigg\{ t + \sum_{i=1}^{j} \sigma_p(\xi_{i}) + \tilde{f}(\xi_j)-\tilde{f}(\xi_0)\geqslant 0, \; k+1 \leqslant j\leqslant n \bigg\}. \end{align}\] By 126 , it holds that \(\{\tilde{\tau}^{\mathfrak f}_{t}>n\} =\{ \tilde{\tau}^{\mathfrak f}_{t} >k\} \bigcap A_{k,n}(t)\). Hence, by taking the conditional expectation with respect to \(\mathscr G_k\), we get that for any \(a\in \mathbb{A}_p\), \(n > k \geqslant 1\) and \(t \in \mathbb{R}\), \[\begin{align} \label{PPPPR001} \mathbb{E}_a \bigg( t + \sum_{i=1}^{n+p} \sigma_p(\xi_i); \tilde{\tau}^{\mathfrak f}_{t} > n \bigg | \mathscr G_k \bigg) = \mathbb{1}_{\{\tilde{\tau}^{\mathfrak f}_{t} > k\} } \mathbb{E}_a \bigg( t + \sum_{i=1}^{n+p} \sigma_p(\xi_i); A_{k,n}(t) \bigg | \mathscr G_k \bigg). \end{align}\tag{129}\] We shall use the following conditioning formula: for any bounded Borel measurable function \(h: \mathbb{A}_p ^{n+p+1} \to \mathbb{R}\) and \(\eta_0,\ldots,\eta_{k} \in \mathbb{A}_p\), denote by \(h_{\eta_0,\ldots,\eta_{k}}\) the bounded Borel measurable function on \({\mathbb{A}}_p^{n+p-k+1}\) defined by \(h_{\eta_0,\ldots,\eta_{k}} (\xi_0,\ldots,\xi_{n+p-k})= h(\eta_0,\ldots,\eta_{k},\xi_1,\ldots,\xi_{n+p-k}),\) then it holds \[\begin{align} \label{PPPPR002} \mathbb{E}_a (h(\xi_0,\ldots,\xi_{n+p} )\big | \xi_0=\eta_0,\ldots,\xi_{k}=\eta_{k} ) = \mathbb{E}_{\eta_{k}} h_{\eta_0,\ldots,\eta_{k}}(\xi_{0},\ldots, \xi_{n+p-k}). \end{align}\tag{130}\] Choose \(\eta_0,\ldots,\eta_{k} \in \mathbb{A}_p\) and set \(t' = t + \sum_{i=1}^{k} \sigma_p(\eta_i) +f(\eta_k)-f(\eta_0)\). Then, by 130 , we get \[\begin{align} \label{PPPPR003} &\mathbb{E}_a \bigg( t + \sum_{i=1}^{n+p} \sigma_p(\xi_i); A_{k,n}(t) \bigg | \xi_0=\eta_0,\ldots,\xi_{k}=\eta_{k} \bigg) \notag \\ &= \mathbb{E}_a \bigg( t + \sum_{i=1}^{k} \sigma_p(\xi_i) + \sum_{i=k+1}^{n+p} \sigma_p(\xi_i); A_{k,n}(t) \bigg | \xi_0=\eta_0,\ldots,\xi_{k}=\eta_{k} \bigg) \notag \\ &= \mathbb{E}_{\eta_{k}} \bigg( t' + \sum_{i=1}^{n+p-k} \sigma_p(\xi_{i}); \tilde{\tau}^{\mathfrak f}_{t'} > n-k \bigg) = W^{\mathfrak f}_{n-k} (\eta_{k}, t'). \end{align}\tag{131}\] Since \(\{ \tilde{\tau}^{\mathfrak f}_{t} >k\}\) and \(B_k\) are \(\mathscr G_k\)-measurable, the assertion of the lemma now follows from 129 and 131 . ◻

Proof. We fix \(a=(g_0,\ldots,g_{p},x,q) \in \mathbb{A}_p\) and \(t\in \mathbb{R}\). Set \(x'=g_p\cdots g_1x\) and \(t' = t + \sigma(g_p\cdots g_1, x) - \tilde{f}(a)\). Consider the event \[\begin{align} B_{n}=\left\{ \max_{p < k\leqslant n}\left\vert \tilde{f}(\xi_k) \right\vert \leqslant c_1 \log n \right\}, \end{align}\] where the constant \(c_1>0\) will be chosen later. Then \[\begin{align} \label{decom-probab-tauft} \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} \geqslant n \right) \leqslant\mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} \geqslant n, B_n \right) + \mathbb{P}_a \left( B_n^c \right). \end{align}\tag{136}\] By Proposition 11, there exist constants \(c, \beta>0\) such that for any \(n \geqslant 1\), \(x \in \mathbb{X}\) and \(t \in \mathbb{R}\), \[\begin{align} \mathbb{P} (\vartheta_{x,t} \geqslant n) \leqslant c \frac{1+\max \{t,0\}}{n^{\beta}}, \end{align}\] where \(\vartheta_{x,t}\) is a stopping time defined by 93 . Using this bound, for the first term in 136 , we get \[\begin{align} \label{probab-tauft-Bn} \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} \geqslant n, B_n \right) &\leqslant\mathbb{P} \left( \vartheta_{x' , t' + c_1 \log n} \geqslant n - p \right) \notag\\ &\leqslant c \frac{\max \{t,0\} + | \sigma(g_p\cdots g_1, x) | +|\tilde{f}(a)| + \log n}{(n -p)^{\beta}}, \end{align}\tag{137}\] with \(c>0\) not depending on \(\tilde{f}\). For the second term in 136 , by Markov’s inequality and 125 , we obtain \[\begin{align} \label{PAbar32bound} \mathbb{P}_a\left( B_n^c\right) \leqslant \sum_{k = p+1}^n \mathbb{P}_a\left( \left\vert \tilde{f}(\xi_k)\right\vert > c_1 \log n \right) \leqslant\frac{c}{n^{\alpha c_1}} \sum_{k = p+1}^n \mathcal{F}_k(a) \leqslant\frac{c }{n^8} \sum_{k = p+1}^n \mathcal{F}_k(a), \end{align}\tag{138}\] where in the last inequality we take \(c_1>0\) to be sufficiently large. Combining the bounds 136 , 137 and 138 completes the proof of the lemma. ◻

Proof. By construction, we have \[\begin{align} \int_{\mathbb{X}} \mathbb{P} \left( \tau^{\mathfrak f}_{x, t} > n \right) \nu(dx) = \int_{\mathbb{X}} \int _{\mathbb{G}^{\{0,\ldots, p\}} } \mathbb{P}_{(g_0,\ldots, g_p, x, 0)} \left( \tilde{\tau}^{\mathfrak f}_{t} > n \right) \mu(dg_0) \ldots \mu(dg_p) \nu(dx). \end{align}\] The conclusion follows from Lemma 15 since, for every \(k \in \mathbb{N}\), we have \[\begin{align} \label{stationary-nu-001} & \int_{\mathbb{X}} \int_{\mathbb{G}^{p+1} } \mathcal{F}_k(g_0,\ldots, g_p, x, 0) \mu(dg_0) \ldots \mu(dg_p) \nu(dx) \notag\\ & = \int_{\mathbb{X}} \int_{\mathbb{G}^k} \int_{\Omega} e^{\alpha |f_k(\omega, g_k \cdots g_1 x)|} \mathbb{P}(d\omega) \mu(dg_1) \ldots \mu(dg_k) \nu(dx) \notag\\ & = \int_{\mathbb{X}} \int_{\Omega} e^{\alpha |f_k(\omega, x)|} \mathbb{P}(d\omega) \nu(dx) \leqslant C_{\alpha}(\mathfrak f), \end{align}\tag{139}\] where we have used the fact that the measure \(\nu\) is \(\mu\)-stationary and \(C_{\alpha}(\mathfrak f)\) is the finite constant appearing in 76 . ◻

Proof. By Hölder’s inequality and Lemma 15, we get \[\begin{align} J & \leqslant\bigg[ \mathbb{E}_a^{1/2} \bigg( \bigg|\sum_{i=n+1}^{n+p} \sigma_p(\xi_i) \bigg|^2 \bigg) + \mathbb{E}_a^{1/2}\left( |\tilde{f} (\xi_n)|^2 \right) +|\tilde{f}(\xi_0)| \bigg] \mathbb{P}_a^{1/2} \left(\tilde{\tau}^{\mathfrak f}_{t} > n\right) \notag\\ & \leqslant c\left( \sqrt{p} + \mathcal{F}_n(a)^{1/2} + |\tilde{f}(a)| \right) \notag\\ & \qquad \times \left[\frac{\max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)|+ \log n}{(n-p)^{3\beta}} + \frac{ \sum_{k = p+1}^n \mathcal{F}_k(a) }{n^8} \right]^{1/2}, \end{align}\] from which the assertion follows immediately. ◻

Proof. Similarly to 127 , by construction we have \[\begin{align} I_2 & = \int_{\mathbb{X}} \int _{\mathbb{G}^{\{0,\ldots, p\}} } \mathbb{E}_{(g_0,\ldots, g_p, x, 0)} \bigg( -\sum_{i=n+1}^{n+p} \sigma_p(\xi_i) + \tilde{f}(\xi_n) - \tilde{f}(\xi_0); \tilde{\tau}^{\mathfrak f}_{t} > n \bigg) \notag\\ & \qquad\qquad \mu(dg_0) \ldots \mu(dg_p) \nu(dx). \end{align}\] The expectation inside the above integral is bounded by using Lemma 16 with \(\alpha/2\) instead of \(\alpha\) in \(\mathcal{F}_k\). The conclusion now follows by using 139 . ◻

Proof. Consider the event \[\begin{align} A_{n}=\left\{ \max_{p < k \leqslant n}\left\vert h_p(\xi_k) \right\vert \leqslant\frac{1}{p + 3} n^{1/2-2\varepsilon}, \max_{p < k \leqslant n}\left\vert \tilde{f}(\xi_k) \right\vert \leqslant\frac{1}{p + 3} n^{1/2-2\varepsilon} \right\}. \end{align}\] We denote \(t' = t + \sigma(g_p\cdots g_1,x)\) and write \[\begin{align} \label{eq-lemma1-000} \mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \bigg\vert; \; p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n \bigg) & = \mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \bigg\vert; \; p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n, A_{n} \bigg) \notag \\ & \quad + \mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \bigg\vert; \; p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n, A_n^c \bigg). \end{align}\tag{140}\] For the first term, on the event \(\{p < \tilde{\tau}^{\mathfrak f}_{t}\}\), since \(\tilde{\tau}^{\mathfrak f}_{t}\) is defined as the first integer \(k > p\) when \(t' + \sum_{i=1}^{k -p} h_p(\xi_i) + \tilde{f}(\xi_{k}) - \tilde{f}(\xi_0)\) becomes negative (see 126 ), and the size of the jump is bounded by \(|h_p(\xi_{\tilde{\tau}^{\mathfrak f}_{t} -p})| + |\tilde{f}(\xi_{\tilde{\tau}^{\mathfrak f}_{t}})| + |\tilde{f}(\xi_{\tilde{\tau}^{\mathfrak f}_{t}-1})|\) which in turn does not exceed \(\frac{3}{p +3}n^{1/2-2\varepsilon}\) on the event \(A_{n}\), it follows that \[\begin{align} \label{eq-lemma1-R1} \mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \bigg\vert; \; p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n, A_{n} \bigg) & \leqslant\mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{\tilde{\tau}^{\mathfrak f}_{t} -p} h_p(\xi_i) \bigg\vert; \; p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n, A_{n} \bigg) \notag\\ & \quad + \mathbb{E}_a \bigg( \bigg\vert \sum_{i=\tilde{\tau}^{\mathfrak f}_{t} -p + 1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \bigg\vert;\; p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n, A_{n} \bigg) \notag\\ & \leqslant\frac{3}{p+3} n^{1/2-2\varepsilon} + \frac{p}{p +3} n^{1/2-2\varepsilon} \notag\\ & \leqslant n^{1/2-2\varepsilon} \leqslant{t\over n^{\varepsilon}}, \end{align}\tag{141}\] where the last inequality holds since \(t\geqslant n^{1/2-\varepsilon}\).

We proceed to give an upper bound for the second term on the right-hand side of 140 . By Markov’s inequality, the exponential moment assumption 71 and the fact that \(\mathcal{F}_k(a) \geqslant 1\) (see 125 ), we get \[\begin{align} \mathbb{P}_a (A_n^c) & \leqslant \mathbb{P}_a\left( \max_{p < k\leqslant n} |h_p(\xi_k)| > \frac{1}{p+3} n^{1/2-2\varepsilon} \right) + \mathbb{P}_a\left( \max_{p < k\leqslant n} |\tilde{f}(\xi_{k})| > \frac{1}{p+3} n^{1/2-2\varepsilon} \right) \notag \\ & \leqslant (n-p) \sup_{x\in \mathbb{X}} \mathbb{P}\left( |\sigma(g_{1}, x)| > \frac{1}{p+3} n^{1/2-2\varepsilon} \right) + \sum_{k = p+1}^n \mathbb{P}_a \left( |\tilde{f}(\xi_k)| > \frac{1}{p+3} n^{1/2-2\varepsilon} \right) \notag \\ & \leqslant c (n-p) e^{- \frac{\alpha}{p+3} n^{1/2-2\varepsilon} } + e^{- \frac{\alpha}{p+3} n^{ 1/2 -2\varepsilon}} \sum_{k = p+1}^n \mathcal{F}_k(a) \notag \\ & \leqslant c e^{- \frac{\alpha}{p+3} n^{ 1/2 - 2\varepsilon}} \sum_{k = p+1}^n \mathcal{F}_k(a). \end{align}\] Therefore, by Hölder’s inequality, \[\begin{align} \label{eq-lemma1-R2} \mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \bigg\vert; \; p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n, A_n^c \bigg) & = \sum_{k =p+ 1}^n \mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \bigg\vert; \; \tilde{\tau}^{\mathfrak f}_{t} = k, A_n^c \bigg) \notag\\ & \leqslant\sum_{k =p+ 1}^n \mathbb{E}_a \bigg( \bigg\vert t' + \sum_{i=1}^{k} h_p(\xi_i) \bigg\vert; \; A_n^c \bigg) \notag\\ & \leqslant\sum_{k = p+1}^n \mathbb{E}_a^{1/2} \bigg( \bigg\vert t' + \sum_{i=1}^{k} h_p(\xi_i) \bigg\vert^2 \bigg) \mathbb{P}_a^{1/2}(A_n^c) \notag\\ & \leqslant c n (|t'|+ n^{1/2}) e^{- \frac{\alpha}{2(p+3)} n^{1/2-2\varepsilon} } \sum_{k = p+1}^n \mathcal{F}_k(a), \end{align}\tag{142}\] where in the last inequality we used the bound \[\begin{align} \label{Inequ-Minkowski-01} \mathbb{E}_a^{1/2} \bigg( \bigg\vert t' + \sum_{i=1}^{k} h_p(\xi_i) \bigg\vert^2 \bigg) \leqslant|t'|+ \sup_{x\in \mathbb{X}} \sqrt{\mathbb{E} (S_{k}^{x} )^{ 2 } } \leqslant|t'|+c n^{1/2}, \end{align}\tag{143}\] which holds by Minkowski’s inequality. Combining 140 , 141 and 142 completes the proof of the lemma. ◻

Proof. For \(x\in \mathbb{X}\) and \(n \geqslant 1\), consider the event \[\begin{align} B_{n,x}=\left\{ \omega : \max_{0 \leqslant k\leqslant n}\left\vert f_k\circ T^k (\omega,x) - f_{k,p}\circ T^k(\omega,x) \right\vert \leqslant\frac{1}{2}n^{-\gamma} \right\}. \end{align}\] Since \[\begin{align} & \mathbb{E} \left( t + S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x} \right) \notag\\ & \leqslant\mathbb{E} \left( t + n^{-\gamma} + S_n^x + f_{n,p} \circ T^n(\omega,x) - f_{0, p}(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x} \right) \notag\\ & \leqslant\mathbb{E} \left( t + n^{-\gamma} + S_n^x + f_{n,p} \circ T^n(\omega,x) - f_{0, p}(\omega,x); \tau^{\mathfrak f_p}_{x, t+n^{-\gamma}} > n \right), \end{align}\] where \(\tau^{\mathfrak f_p}_{x, t+n^{-\gamma}}\) is defined by 144 , we get \[\begin{align} &\mathbb{E} \left( t + S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n \right) \notag\\ &= \mathbb{E} \left( t + S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x} \right) \notag\\ &\qquad\qquad\qquad + \mathbb{E} \left( t + S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x}^c \right) \notag\\ &\leqslant\mathbb{E} \left( t + n^{-\gamma} + S_n^x + f_{n,p} \circ T^n(\omega,x) - f_{0, p}(\omega,x); \tau^{\mathfrak f_p}_{x, t+n^{-\gamma}} > n \right) \notag\\ &\qquad\qquad\qquad + \mathbb{E} \left( t + S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x}^c \right). \end{align}\] By integrating over \(x\in\mathbb{X}\), and using 81 and 145 , we obtain \[\begin{align} U_n^{\mathfrak f}(t) \leqslant U_n^{\mathfrak f_p}(t+n^{-\gamma}) + \int_{\mathbb{X}} \mathbb{E} \left( t + S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x}^c \right) \nu(dx). \end{align}\] By the Cauchy-Schwarz inequality, we see that \[\begin{align} &\int_{\mathbb{X}} \mathbb{E} \left( t + S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x}^c \right) \nu(dx) \notag\\ &= t \int_{\mathbb{X}} \mathbb{P} \left( \tau^{\mathfrak f}_{x, t}>n, B_{n,x}^c \right) \nu(dx) \notag\\ & \quad + \int_{\mathbb{X}} \mathbb{E} \left( S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x); \tau^{\mathfrak f}_{x, t} > n, B_{n,x}^c \right) \nu(dx) \notag\\ &\leqslant\max \{t,0\} \int_{\mathbb{X}} \mathbb{P} ( B_{n,x}^c ) \nu(dx) \notag\\ & \quad + \left(\int_{\mathbb{X}} \mathbb{E} \left( \big| S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x) \big|^2 \right) \nu(dx) \right)^{1/2} \left(\int_{\mathbb{X}} \mathbb{P} (B_{n,x}^c) \nu(dx)\right)^{1/2}. \end{align}\] By the definition of \(T\) on the space \(\Omega\times \mathbb{X}\) (cf.@eq:def-T-Omega-X ), the \(\mu\)-stationarity of the measure \(\nu\), Chebyshev’s inequality and the approximation property 78 , we obtain \[\begin{align} \label{proba-Bnx-001} \int_{\mathbb{X}} \mathbb{P}(B_{n,x}^c) \nu(dx) &\leqslant\sum_{k=0}^n \int_{\mathbb{X}} \mathbb{P}\left( e^{\alpha \left| f_k \circ T^k(\omega, x) - f_{k,p} \circ T^k(\omega, x) \right| } - 1 \geqslant e^{\frac{\alpha}{2} n^{-\gamma}}-1\right) \nu(dx) \notag\\ &= \sum_{k=0}^n \int_{\mathbb{X}} \mathbb{P}\left( e^{\alpha \left\vert f_k(\omega, x) - f_{k,p}(\omega, x) \right\vert } - 1 \geqslant e^{\frac{\alpha}{2} n^{-\gamma}}-1\right) \nu(dx) \notag\\ &\leqslant\frac{1}{e^{\frac{\alpha}{2} n^{-\gamma}}-1} \sum_{k=0}^n \int_{\mathbb{X}} \mathbb{E}\left( e^{\alpha \left\vert f_k(\omega, x) - \mathbb{E}( f_{k}(\cdot, x) | \mathscr{A}_p)(\omega) \right\vert } - 1 \right) \nu(dx) \notag\\ &\leqslant c n^{1+\gamma} e^{ -\beta p} D_{\alpha,\beta}(\mathfrak f) \leqslant c n^{1+\gamma} e^{ -\beta n^{\delta}} D_{\alpha,\beta}(\mathfrak f). \end{align}\tag{146}\] Using Minkowski’s inequality and the assumptions 71 , 72 and 76 , we get \[\begin{align} \label{bound-Snx-fn-001} & \left(\int_{\mathbb{X}} \mathbb{E}\left( \big| S_n^x + f_n \circ T^n (\omega,x)-f_0(\omega,x) \big|^2 \right) \nu(dx) \right)^{1/2} \notag\\ &\leqslant\left(\int_{\mathbb{X}} \mathbb{E} \big( |S_n^x|^2 \big) \nu(dx) \right)^{1/2} + \left(\int_{\mathbb{X}} \mathbb{E}\left( \big| f_n \circ T^n (\omega,x)-f_0(\omega,x) \big|^2 \right) \nu(dx) \right)^{1/2} \notag\\ &\leqslant c \left( n + C_{\alpha}(\mathfrak f) \right). \end{align}\tag{147}\] Combining these bounds, we derive the upper bound in ?? . The lower bound can be established similarly. ◻

Proof. We use the notation from the proof of Proposition 14 and we write \[\begin{align} \int_{\mathbb{X}} \mathbb{P} \left( \tau^{\mathfrak f}_{x, t} > n \right) \nu(dx) & = \int_{\mathbb{X}} \mathbb{P} \left( \tau^{\mathfrak f}_{x, t} > n, B_{n,x} \right) \nu(dx) + \int_{\mathbb{X}} \mathbb{P} \left( \tau^{\mathfrak f}_{x, t} > n, B_{n,x}^c \right) \nu(dx) \notag\\ & \leqslant\int_{\mathbb{X}} \mathbb{P} \left( \tau^{\mathfrak f_p}_{x, t + n^{-\gamma}} > n \right) \nu(dx) + \int_{\mathbb{X}} \mathbb{P} \left( B_{n,x}^c \right) \nu(dx). \end{align}\] By Corollary 11, the first term is bounded by \(c \frac{ \max \{t,0\} +1 }{n^{\varepsilon}} C_{\alpha}(\mathfrak f)\). By 146 , the second term is dominated by \(c n^{1+\gamma} e^{ -\beta n^{\delta}} D_{\alpha,\beta}(\mathfrak f)\). Combining these bounds gives the desired result. ◻

Proof. Using the Cauchy-Schwarz inequality, Lemma 10, the bounds 147 and 84 , we get that for any \(n\geqslant 1\) and \(p\geqslant n^\delta\), \[\begin{align} &\left| U^{\mathfrak f,\theta}_n (t)-U^{\mathfrak f, \theta_p}_n (t) \right| \\ &\leqslant\int_{\mathbb{X}} \mathbb{E} \left(\left| t + S^{x}_n + f_n \circ T^n (\omega,x)- f_0(\omega,x) \right| \left| \theta - \theta_p \right|; \tau^{\mathfrak f}_{x, t} > n \right) \nu(dx) \\ &\leqslant\left( \int_{\mathbb{X}} \mathbb{E} \left( \left| t + S^{x}_n + f_n \circ T^n (\omega,x)- f_0(\omega,x) \right|^2 ; \tau^{\mathfrak f}_{x, t} > n \right) \nu(dx) \right)^{1/2} \mathbb{E}^{1/2} \left(\theta - \theta_p \right)^2 \\ &\leqslant c \left( \max \{t, 0\} + n + B^{1/2} \right) \| \theta - \theta_p \|_{\infty}^{1/2} \; \mathbb{E}^{1/2} |\theta - \theta_p | \\ &\leqslant c \left( \max \{t, 0\} + n + B \right) B e^{-\beta n^{\delta}/2} \\ &\leqslant c' e^{-\beta n^{\delta}/4} \left(1+ \max \{t, 0\} \right). \end{align}\] By applying the same techniques used in the proof of Proposition 14, we obtain ?? . ◻

Proof. Let \(\beta\) be as in Lemma 15 and choose \(\gamma > 8/\alpha\). Note that, by 125 , for any \(k > p\) and \(a \in \mathbb{A}_p\), we have \[\begin{align} \label{BOUND32XI-002-001} \mathbb{E}_a |\tilde{f}(\xi_k)| \leqslant c \mathcal{F}_k(a). \end{align}\tag{148}\] Now we handle the term involving \(\tilde{f}\). Note that \[\begin{align} \label{HHH-0001-001-OO2} \mathbb{E}_a \left( |\tilde{f}(\xi_{\tilde{\tau}^{\mathfrak f}_{t}})| ; \tilde{\tau}^{\mathfrak f}_{t} > n \right) & = \sum_{k=n+1}^{\infty} \mathbb{E}_a \left( |\tilde{f}(\xi_{\tilde{\tau}^{\mathfrak f}_{t}})|; \tilde{\tau}^{\mathfrak f}_{t} = k \right) \notag\\ & = \sum_{k=n+1}^{\infty} \mathbb{E}_a \left(|\tilde{f}(\xi_{k})|; |\tilde{f}(\xi_{k})| > \gamma \log k, \tilde{\tau}^{\mathfrak f}_{t} = k \right) \notag\\ & \quad + \sum_{k=n+1}^{\infty} \mathbb{E}_a \left( |\tilde{f}(\xi_{k})|; |\tilde{f}(\xi_{k})| \leqslant\gamma \log k , \tilde{\tau}^{\mathfrak f}_{t} = k \right). \end{align}\tag{149}\] For the first term in 149 , by Chebyshev’s inequality, 148 and 125 , we get \[\begin{align} \mathbb{E}_a \left(|\tilde{f}(\xi_{k})|; |\tilde{f}(\xi_{k})| > \gamma \log k, \tilde{\tau}^{\mathfrak f}_{t} = k \right) &\leqslant\mathbb{E}_a \left(|\tilde{f}(\xi_{k})|; |\tilde{f}(\xi_{k})| > \gamma \log k \right) \\ & \leqslant k^{-\alpha \gamma/2} \mathbb{E}_a \left( |\tilde{f}(\xi_{k})| e^{ \frac{\alpha}{2} |\tilde{f}(\xi_{k})| } \right) \\ & \leqslant c k^{-\alpha \gamma/2} \mathbb{E}_a e^{ \alpha |\tilde{f}(\xi_{k})| } \\ & = c k^{-\alpha \gamma/2} \mathcal{F}_k(a). \end{align}\] As \(\gamma > 8/\alpha\), summing over \(k\) gives the following bound for the first term on the right-hand side of 149 : \[\begin{align} \label{BOUND32XI-002-002} \sum_{k=n+1}^{\infty} \mathbb{E}_a \left(|\tilde{f}(\xi_{k})|; |\tilde{f}(\xi_{k})| > \gamma \log k, \tilde{\tau}^{\mathfrak f}_{t} = k \right) & \leqslant c n^{2-\alpha \gamma/2} \sum_{k=n+1}^{\infty} \frac{1}{k^2} \mathcal{F}_k(a) \notag\\ & \leqslant c n^{-2} \sum_{k=n+1}^{\infty} \frac{1}{k^2} \mathcal{F}_k(a). \end{align}\tag{150}\] For the second term on the right-hand side of 149 , we have \[\begin{align} \label{HHH-0001-002-O2} \sum_{k=n+1}^{\infty} \mathbb{E}_a \left( |\tilde{f}(\xi_{k})|; |\tilde{f}(\xi_{k})| \leqslant\gamma \log k , \tilde{\tau}^{\mathfrak f}_{t} = k \right) & \leqslant\gamma \sum_{k=n+1}^{\infty} \log k \;\mathbb{P}_a \big( \tilde{\tau}^{\mathfrak f}_{t} = k \big) \notag\\ &= \gamma \mathbb{E}_a \left( \log \tilde{\tau}^{\mathfrak f}_{t}; \tilde{\tau}^{\mathfrak f}_{t} >n \right). \end{align}\tag{151}\] Recall that, for a random variable \(Y\) without atoms and a smooth and integrable function \(\phi\) on \(\mathbb{R}\), and any \(t \in \mathbb{R}\), we have \[\begin{align} \mathbb{E} \left( \phi (Y) \mathbb{1}_{ \{ Y > t \} } \right) = \phi(t) \mathbb{P} (Y > t) + \int_t^{\infty} \phi'(u) \mathbb{P} (X > u) du. \end{align}\] Applying this identity gives \[\begin{align} \mathbb{E}_a \left( \log \tilde{\tau}^{\mathfrak f}_{t}; \tilde{\tau}^{\mathfrak f}_{t} >n \right) = (\log n) \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} > n \right) + \int_{n}^{\infty} u^{-1} \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} > u \right) du. \end{align}\] Hence, using Lemma 15, we get \[\begin{align} \label{BOUND32XI-002-003} &\mathbb{E}_a \left( \log \tilde{\tau}^{\mathfrak f}_{t}; \tilde{\tau}^{\mathfrak f}_{t} >n \right) \notag\\ &\leqslant (\log n) \bigg( c \frac{\max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)|+ \log n}{(n-p)^{\beta}} + \frac{c }{n^8} \sum_{k = p+1}^{n} \mathcal{F}_k(a) \bigg) \notag\\ & \quad + \int_{n+1}^{\infty} u^{-1} \bigg( c \frac{\max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)|+ \log u}{(u-p)^{\beta}} + \frac{c }{u^8} \sum_{k = p+1}^{[u]} \mathcal{F}_k(a) \bigg) du \notag\\ & \leqslant c \left( \max \{t,0\} + |\sigma(g_p\cdots g_1,x)| + |\tilde{f}(a)| \right) (n-p)^{-\beta} + c n^{-2} \sum_{k=p+1}^{\infty} \frac{1}{k^2} \mathcal{F}_k(a). \end{align}\tag{152}\] Combining 149 , 150 , 151 and 152 gives \[\begin{align} \label{HHH-0001-005-OO2} & \mathbb{E}_a \left( |\tilde{f}(\xi_{\tilde{\tau}^{\mathfrak f}_{t}})| ; \tilde{\tau}^{\mathfrak f}_{t} > n \right) \notag \\ &\leqslant c \left( \max \{t,0\} + |\sigma(g_p\cdots g_1,x)| + |\tilde{f}(a)| \right) (n-p)^{-\beta} + c n^{-2} \sum_{k=p+1}^{\infty} \frac{1}{k^2} \mathcal{F}_k(a). \end{align}\tag{153}\] The term involving \(h_p\) is dominated in a similar manner. ◻

Proof. Let \(a=(g_{0},\ldots,g_{p},x,q) \in \mathbb{A}_p\) and \(t'=t+ \sigma(g_{p} \cdots g_{1}, x).\) In view of Lemma 15, to obtain inequality 154 , it is sufficient to prove the result with \(W^{\mathfrak f}_n(a, t)\) replaced by \(\mathbb{E}_a ( t'+M_{n} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > n )\). We first show that for any \(1 \leqslant p \leqslant n \leqslant m\), \(a \in \mathbb{A}_p\) and \(t \in \mathbb{R}\), \[\begin{align} \label{Equ-desired-O2} & \mathbb{E}_a\left( t'+M_{n}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > n \right) \leqslant\mathbb{E}_a\left( t'+M_{ m}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > m\right) \notag\\ & \qquad\qquad\qquad\qquad + \mathbb{E}_a \Bigg( \sum_{j=\tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_j) - \tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} }); \; n + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant m \Bigg). \end{align}\tag{155}\] Indeed, since \(\mathbb{E}_a (M_{m}) = \mathbb{E}_a (M_{ n})\), it holds that \[\begin{align} & \mathbb{E}_a \left( t'+M_{m}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant m \right) - \mathbb{E}_a \left( t'+M_{n}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant n \right) \notag\\ & = \mathbb{E}_a \left( t'+M_{n}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > n \right) - \mathbb{E}_a\left( t'+M_{ m}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > m \right), \end{align}\] so that 155 is equivalent to the following inequality: \[\begin{align} \label{Equ-00a-O2} & \mathbb{E}_a\left( t'+M_{m}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant m \right) \leqslant\mathbb{E}_a\left( t'+M_{n}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant n \right) \notag\\ & \qquad\qquad\qquad\qquad + \mathbb{E}_a \Bigg( \sum_{j=\tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_j) - \tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} });\; n + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant m \Bigg). \end{align}\tag{156}\] We shall prove 156 using an induction argument. Let \(k \in [n + 1, m]\) and we write \[\begin{align} \label{E-Mh-tau-ft-001} & \mathbb{E}_a \left( t' + M_k - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k \right) \notag\\ & = \mathbb{E}_a \left( t' + M_k - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k - 1 \right) + \mathbb{E}_a \left( t' + M_k - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} = k \right). \end{align}\tag{157}\] As the event \(\{ \tilde{\tau}^{\mathfrak f}_{t} \leqslant k -1 \}\) is in \(\mathscr G_{k-1}\), by the martingale property, we get \[\begin{align} \label{E-Mh-tau-ft-002} \mathbb{E}_a \left( t' + M_k - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k -1 \right) = \mathbb{E}_a \left( t' + M_{k -1} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k -1\right). \end{align}\tag{158}\] Besides, by the definition of \(\tilde{\tau}^{\mathfrak f}_{t}\) and \(M_n\) (cf.@eq:def-tau-f-y and 134 ), on the set \(\{\tilde{\tau}^{\mathfrak f}_{t} = k \}\), we have \(t' + M_{k-p} + \tilde{f}(\xi_k) - \tilde{f}(\xi_0)<0\), which leads to \[\begin{align} \label{E-Mh-tau-ft-003} \mathbb{E}_a \left( t' + M_k - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} = k \right) \leqslant\mathbb{E}_a \Bigg( \sum_{j=k-p+1}^k h_p(\xi_j) - \tilde{f}(\xi_k) ;\;\tilde{\tau}^{\mathfrak f}_{t} = k \Bigg). \end{align}\tag{159}\] Hence, combining 157 , 158 and 159 gives that for any \(k \in [n+1, m]\), \[\begin{align} & \mathbb{E}_a \left( t' + M_k - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k \right) \notag\\ & \leqslant\mathbb{E}_a \left( t'+M_{k -1}-\tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k -1 \right) + \mathbb{E}_a \Bigg( \sum_{j=k-p+1}^k h_p(\xi_j) - \tilde{f}(\xi_k) ;\;\tilde{\tau}^{\mathfrak f}_{t} = k \Bigg). \end{align}\] Summing over \(k \in [n+1, m]\), we get \[\begin{align} & \mathbb{E}_a \left( t'+M_{m}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant m \right) \notag\\ & \leqslant\mathbb{E}_a \left( t'+M_{n}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant n \right) + \sum_{k = n+1}^{ m } \mathbb{E}_a \Bigg( \sum_{j = k - p + 1}^k h_p(\xi_j) - \tilde{f}(\xi_k) ;\;\tilde{\tau}^{\mathfrak f}_{t} = k \Bigg) \notag\\ & = \mathbb{E}_a \left( t'+M_{n}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant n \right) + \mathbb{E}_a \Bigg( \sum_{j=\tilde{\tau}^{\mathfrak f}_{t}-p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_j) - \tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} }); \; n + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant m \Bigg), \end{align}\] thus proving 156 . Since 155 is equivalent to 156 , the conclusion of the lemma follows by Lemma 18. ◻

Proof. This is obtained from Lemma 19 together with 120 , 127 and Corollary 12. Recall that \(\mathbb{E} \sigma(g_{p} \cdots g_{1}, x)^2 \leqslant c p\) by the moment assumption 71 and the centering assumption 72 . Since \(a=(g_0,\ldots,g_{p},x,0) \in \mathbb{A}_p\), integrating both sides of 154 in Lemma 19 (with \(\alpha/4\) instead of \(\alpha\)) with respect to the measure \(\mu(dg_0) \ldots \mu(dg_{p}) \nu(dx)\) and applying 127 yields that for any \(t \in \mathbb{R}\), \(n\geqslant 1\) and \(1\leqslant p \leqslant n^{\varepsilon/2}\), \[\begin{align} \label{Expect-E95x001-002-a} \int_{\mathbb{X}} \mathbb{E} \left( t+S_{n+p}^x ; \tau^{\mathfrak f}_{x, t}>n \right) \nu(dx) & \leqslant \int_{\mathbb{X}} \mathbb{E} \left( t+S_{m+p}^x ; \tau^{\mathfrak f}_{x, t}> m \right) \nu(dx) \notag \\ & \quad + \frac{c_{\varepsilon}}{n^{\varepsilon/2}} \left(1+ \max \{t,0\} \right) C_{\alpha}(\mathfrak f), \end{align}\tag{160}\] where we have used the fact that \(\nu\) is \(\mu\)-stationary, as in 139 . Using Lemma 12, we can slightly modify the expectations in both sides of 160 , leading to \[\begin{align} U_n^{\mathfrak f}(t) &= \int_{\mathbb{X}} \mathbb{E} \left( t+S_{n}^x + f_n( T^n (\omega,x) ) - f_0(\omega,x) ; \tau^{\mathfrak f}_{x, t}>n \right) \nu(dx) \notag \\ & \leqslant U_{m}^{\mathfrak f}(t) + \frac{c_{\varepsilon}}{n^{\varepsilon/2}} \left(1+ \max \{t,0\} \right) C_{\alpha}(\mathfrak f). \end{align}\] This completes the proof of Lemma 20. ◻

Proof. Fix \(k\) to be a large integer whose value will be determined later. Let \(h\) be the least integer with the property that \(h^{k} > n\) and let \(l\) be the least integer with the property that \(l^{k} > m\). We set \(m_{h-1}=n\), \(m_i = i^{k}\) for \(h \leqslant i \leqslant l-1\) and \(m_l = m\). By Lemma 20, there exist constants \(\varepsilon, \beta_0, c>0\) such that, whenever \(p\leqslant m_{i-1}^{\varepsilon}\), the following inequality holds: \[\begin{align} U^{\mathfrak f_{p}}_{m_{i-1}}(t) \leqslant U^{\mathfrak f_{p}}_{m_{i}}(t) + c \left( 1 + \max \{t,0\} \right) m_{i-1}^{-\beta_0} C_{\alpha}(\mathfrak f), \end{align}\] where \(\alpha\) is from the moment condition 76 . In particular, setting \(p_i = [m_{i-1}^{\varepsilon}]\) for \(h\leqslant i\leqslant l\), we get \[\begin{align} \label{-thbound32with32m32for32U-200-001} U^{\mathfrak f_{p_{i}}}_{m_{i-1}}(t) \leqslant U^{\mathfrak f_{p_i}}_{m_{i}}(t) + c \left( 1 + \max \{t,0\} \right) m_{i-1}^{-\beta_0} C_{\alpha}(\mathfrak f). \end{align}\tag{161}\] The approximation property 78 implies that there exists \(\beta_1\) such that \(D_{\alpha,\beta_1}(\mathfrak f) <\infty.\) Since the approximation property also holds with \(\beta=\min\{\beta_0,\beta_1\}\), applying Proposition 14 with \(\delta=\varepsilon\), there exists a constant \(c>0\) such that, for any \(h\leqslant i \leqslant l\) and any \(\gamma>0\), \[\begin{align} \label{BBBB002-001} U^{\mathfrak f}_{m_{i-1}} (t) \leqslant U^{\mathfrak f_{p_i}}_{m_{i-1}} (t+ m_{i-1}^{-\gamma}) + c \left(\max \{t,0\} + C_{\alpha}(\mathfrak f) \right) e^{ -\beta m_{i-1}^{\varepsilon}} D_{\alpha,\beta}(\mathfrak f) \end{align}\tag{162}\] and \[\begin{align} \label{BBBB002-002} U^{\mathfrak f_{p_i}}_{m_{i}} (t) \leqslant U^{\mathfrak f}_{m_{i}} (t+ m_{i}^{-\gamma}) + c \left(\max \{t,0\} + C_{\alpha}(\mathfrak f) \right) e^{ -\beta m_{i}^{\varepsilon}} D_{\alpha,\beta}(\mathfrak f), \end{align}\tag{163}\] Combining inequalities 161 , 162 and 163 , and noting that \(m_{i} \geqslant m_{i-1}\), we get \[\begin{align} U^{\mathfrak f}_{m_{i-1}} (t) \leqslant U^{\mathfrak f}_{m_{i}} (t+ 2 m_{i-1}^{-\gamma}) + c \left(e^{ -\beta m_{i-1}^{\varepsilon}} + m_{i-1}^{-\beta} \right) \left( \max \{t,0\} + C_{\alpha}(\mathfrak f)\right) C_{\alpha}(\mathfrak f) D_{\alpha,\beta}(\mathfrak f). \end{align}\] By consecutively applying these inequalities and using the definition of \(m_i\), we obtain \[\begin{align} U^{\mathfrak f}_{n} (t) \leqslant U^{\mathfrak f}_{m} \bigg(t+ 2 \sum_{i=h-1}^{l-1} i^{-\gamma k} \bigg) + c \left(\max \{t,0\} + C_{\alpha}(\mathfrak f) \right) \sum_{i=h-1}^{l-1}\left(e^{ -\beta i^{\varepsilon k}} + i^{-\beta k}\right) C_{\alpha}(\mathfrak f) D_{\alpha,\beta}(\mathfrak f). \end{align}\] Finally, letting \(\gamma>0\) be arbitrary and choosing \(k>\max \{ \gamma^{-1}, \beta^{-1}\}\), we obtain \[\begin{align} U^{\mathfrak f}_{n} (t) \leqslant U^{\mathfrak f}_{m} \left(t+ c n^{- \gamma + k^{-1}} \right) + c n^{-\beta + k^{-1}} \left( \max \{t,0\} + C_{\alpha}(\mathfrak f) \right) C_{\alpha}(\mathfrak f) D_{\alpha,\beta}(\mathfrak f). \end{align}\] The assertion of the proposition follows. ◻

Proof. We write the following extension of formula eq. ¿eq:Expect-E95x001?

for any \(x \in \mathbb{X}\) and \(t \in \mathbb{R}\), \[\begin{align} \label{integral-formula-app} &\int_{\mathbb{X}} \mathbb{E} \left( \left(t + S^{x}_{n+p}\right) \theta(\omega) ; \tau^{\mathfrak f}_{x, t} > n \right) \nu(dx) \notag\\ & = \int_{\mathbb{X}} \int _{\mathbb{G}^{\{0,\ldots, p\}} } W^{\mathfrak f}_{n} ((g_{0},\ldots,g_{p}, x, 0), t) \theta(g_1,\ldots,g_p) \mu(dg_{0}) \ldots \mu(dg_{p}) \nu(dx), \end{align}\tag{164}\] where \(W^{\mathfrak f}_{n}\) is defined by 128 . Then Lemma 21 is obtained from Lemma 19 in the same way as Lemma 20. ◻

Proof of the first part of Theorem 10. Inequality 85 is the conclusion of Proposition 17. ◻

Proof. Using the same notation as in Remark 19, we have \[\begin{align} \mathbb{P}_a\left( \tilde{\nu}_{n,t}>\ell \right) = \mathbb{P} \left( \nu_{n,x',t'}> \ell-p\right) = \mathbb{P} \bigg( \sup_{ 1\leqslant k \leqslant\ell - p} |t' + S_k^{x'} | \leqslant 2 n^{1/2-\varepsilon} \bigg). \end{align}\] By Lemma 8, there exists a constant \(\beta>0\) such that, for any \(x'\in \mathbb{X}\) and \(t'\in \mathbb{R}\), \[\begin{align} \mathbb{P} \bigg( \sup_{ 1\leqslant k \leqslant\ell -p} |t' + S_k^{x'} | \leqslant 2 n^{1/2-\varepsilon} \bigg) \leqslant 2 \exp \left(-\beta \frac{\ell- p}{4 n^{1-2\varepsilon}}\right). \end{align}\] The assertion follows. ◻

Proof. By Remark 19, it holds that, with \(x'=g_p\cdots g_1 x\) and \(t' = t + \sigma(g_p\cdots g_1, x) - \tilde{f}(a)\), \[\begin{align} \mathbb{P}_a \left( \tilde{\nu}_{n,t} \leqslant n^{1/2-\varepsilon} - t' - p \right) = \mathbb{P} \left( \nu_{n,x',t'} \leqslant n^{1/2-\varepsilon} - t' \right). \end{align}\] Now the assertion follows from Lemma 9. ◻

Proof. Let \(x'=g_p\cdots g_1 x\) and \(t' = t + \sigma(g_p\cdots g_1, x)\), where \(a=(g_0,\ldots, g_{p}, x,q) \in \mathbb{A}_p\) and \(t \in \mathbb{R}\). We also fix \(\varepsilon>0\), with its value to be determined later.

Let us first assume that \(t' - \tilde{f}(a) \geqslant-n^{\varepsilon}\), so that we are able to apply Lemma 23. In this case, since the intersection \(\{ \tilde{\nu}_{n,t} > n^{1/2-\varepsilon} - t' + \tilde{f}(a) \} \cap \{ n^{1/2-\varepsilon} - t' + \tilde{f}(a) > \tilde{\tau}^{\mathfrak f}_{t} \}\) is contained in the set \(\{ \tilde{\tau}^{\mathfrak f}_{t} \leqslant\tilde{\nu}_{n,t} \}\), we have \[\begin{align} \label{a001} \mathbb{P}_a( \tilde{\tau}^{\mathfrak f}_{t} > \tilde{\nu}_{n,t} ) \leqslant\mathbb{P}_a ( \tilde{\nu}_{n,t} \leqslant n^{1/2-\varepsilon} - t' + \tilde{f}(a) ) + \mathbb{P}_a ( n^{1/2-\varepsilon} - t' + \tilde{f}(a) \leqslant\tilde{\tau}^{\mathfrak f}_{t}). \end{align}\tag{166}\] For the first term, using Remark 19 and Lemma 23, we get \[\begin{align} \label{a002} \mathbb{P}_a ( \tilde{\nu}_{n,t} \leqslant n^{1/2-\varepsilon} - t' + \tilde{f}(a) ) = \mathbb{P} ( \nu_{n,x',t'} \leqslant n^{1/2-\varepsilon} - t' + \tilde{f}(a) - p ) \leqslant c \exp \left( -c_{\varepsilon}n^{\varepsilon/2} \right). \end{align}\tag{167}\] Now we handle the second term in 166 . Assume that \(t' - \tilde{f}(a)\in [-n^{\varepsilon}, \frac{1}{2} n^{1/2-\varepsilon}]\). Then, by Lemma 15, there exist constants \(\beta>0\) and \(c>0\) such that, for \(p\leqslant\frac{1}{4}n^{1/2-\varepsilon}\), \[\begin{align} &\mathbb{P}_a ( n^{1/2-\varepsilon} - t' + \tilde{f}(a) \leqslant\tilde{\tau}^{\mathfrak f}_{t}) \leqslant\mathbb{P}_a \bigg( \tilde{\tau}^{\mathfrak f}_{t} \geqslant\frac{1}{2} n^{1/2-\varepsilon} \bigg) \notag\\ &\leqslant c \frac{\max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)| + \log( \frac{1}{2} n^{1/2-\varepsilon}) }{(\frac{1}{2} n^{1/2-\varepsilon}-p)^{\beta}} + \frac{c }{n^{4 - 8\varepsilon}} \sum_{k = p+1}^n \mathcal{F}_k(a) \notag\\ &\leqslant \frac{c \log n}{\left(n^{1/2-\varepsilon}-2p\right)^{\beta}} \left( \max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)| \right) + \frac{c }{n^{4 - 8\varepsilon}} \sum_{k = p+1}^n \mathcal{F}_k(a) \notag\\ &\leqslant \frac{c}{n^{\beta/4 }} \left( \max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)| \right) + \frac{c }{n^{4 - 8\varepsilon}} \sum_{k = p+1}^n \mathcal{F}_k(a). \end{align}\] If \(t' - \tilde{f}(a) > \frac{1}{2} n^{1/2-\varepsilon}\), then we have \[\begin{align} \label{a003} \mathbb{P}_a ( n^{1/2-\varepsilon} - t' + \tilde{f}(a) \leqslant\tilde{\tau}^{\mathfrak f}_{t}) \leqslant 1 \leqslant 2 \frac{ t' - \tilde{f}(a) }{n^{1/2-\varepsilon}} \leqslant 2 \frac{ \max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)| }{n^{1/2-\varepsilon}}. \end{align}\tag{168}\] Therefore, in the case \(t' - \tilde{f}(a) \geqslant-n^{\varepsilon}\), the assertion follows from 166 , 167 and 168 by taking \(\varepsilon\) small enough.

It remains to deal with the case \(t' - \tilde{f}(a) < -n^{\varepsilon}\). In this case, we set \[\begin{align} A = \left\{ \xi\in \Omega_p : |\tilde{f}(\xi_{p+1})| \leqslant\frac{n^{\varepsilon}}{2}, \; |\sigma_p\left(\xi_{p+1} \right)| \leqslant\frac{n^{\varepsilon}}{2} \right\}. \end{align}\] In view of the definition 122 of the transition kernel \(\{ P_a: a \in \mathbb{A}_p \}\), we have, for \(p\leqslant n^{\varepsilon/2}\), \[\begin{align} \label{aboundAc001} \mathbb{P}_a\left(A^c\right) &\leqslant\mathbb{P}_a \left( |\tilde{f}(\xi_{p+1})| > \frac{n^{\varepsilon}}{2} \right) + \sup_{x\in \mathbb{X}} \mathbb{P} \left(|\sigma (g_1,x)| \geqslant\frac{n^{\varepsilon}}{2} \right) \notag\\ &\leqslant e^{- \alpha \frac{n^{\varepsilon}}{2}} \mathcal{F}_{p+1}(a) + c e^{- \alpha \frac{n^{\varepsilon}}{2}} \leqslant c' e^{- \alpha \frac{n^{\varepsilon}}{2}} \mathcal{F}_{p+1}(a), \end{align}\tag{169}\] where we have used 71 . On the set \(A\), it holds, \(\mathbb{P}_a\)-almost surely, \[\begin{align} t +\sum_{i=1}^{p+1} \sigma_p (\xi_i) + \tilde{f}(\xi_{p+1}) - \tilde{f}(\xi_0) = t' - \tilde{f}(a)+ \sigma_p (\xi_{p+1}) + \tilde{f}(\xi_{p+1}) < -n^{\varepsilon} + n^{\varepsilon} =0. \end{align}\] Hence, by the definitions 126 and 165 of the stopping times \(\tilde{\tau}_t^{\mathfrak f}\) and \(\tilde{\nu}_{n,t}\), it holds that \(\tilde{\tau}_t^{\mathfrak f} \leqslant p+1 \leqslant\tilde{\nu}_{n,t}\), \(\mathbb{P}_a\)-almost surely on the set \(A\). Therefore, when \(t' - \tilde{f}(a) < -n^{\varepsilon}\), we obtain \[\begin{align} \mathbb{P}_a \left( \tilde{\nu}_{n,t} < \tilde{\tau}_t^{\mathfrak f} \right) \leqslant\mathbb{P}_a \left( A^c \right), \end{align}\] and the conclusion follows from 169 . ◻

Proof. We first prove 170 . By Hölder’s inequality, with \(\eta >1\) and \(\frac{1}{\eta} +\frac{1}{\eta'}=1\), \[\begin{align} \label{app-lem-10-001} & \mathbb{E}_a \Bigg( \Bigg| \sum_{i=\tilde{\nu}_{n,t}-p+1}^{\tilde{\nu}_{n,t}} h_p(\xi_i) \Bigg|; \; \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t}, \tilde{\nu}_{n,t} \leqslant n \Bigg) \notag\\ & \leqslant \mathbb{E}_a^{1/\eta} \Bigg( \Bigg| \sum_{i=\tilde{\nu}_{n,t}-p+1}^{\tilde{\nu}_{n,t}} h_p(\xi_i) \Bigg|^\eta; \tilde{\nu}_{n,t} \leqslant n \Bigg) \mathbb{P}_a^{1/\eta'} \left( \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t} \right). \end{align}\tag{172}\] For the first expectation on the right-hand side of 172 , in view of 165 , we write \[\begin{align} \mathbb{E}_a \Bigg( \Bigg| \sum_{i=\tilde{\nu}_{n,t}-p+1}^{\tilde{\nu}_{n,t}} h_p(\xi_i) \Bigg|^\eta; \tilde{\nu}_{n,t} \leqslant n \Bigg) = \sum_{\ell =p+1}^{n} \mathbb{E}_a \Bigg( \Bigg| \sum_{i=\ell-p+1}^{\ell} h_p(\xi_i) \Bigg|^\eta; \tilde{\nu}_{n,t}=\ell \Bigg) \leqslant c p^{\eta} n, \end{align}\] where the last inequality holds due to the moment assumption 71 . Substituting this into 172 and using Lemma 24, we get that there exist constants \(c, \beta, \varepsilon> 0\) such that, for any \(a \in \mathbb{A}_p\), \(t \in \mathbb{R}\), \(n\geqslant 1\) and \(p\leqslant n^{\varepsilon}\), \[\begin{align} & \mathbb{E}_a \Bigg( \Bigg| \sum_{i=\tilde{\nu}_{n,t}-p+1}^{\tilde{\nu}_{n,t}} h_p(\xi_i) \Bigg|; \; \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t}, \tilde{\nu}_{n,t} \leqslant n \Bigg)\\ &\leqslant c \left( p^{\eta} n \right)^{1/\eta} \mathbb{P}_a^{1/\eta'} \left( \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t} \right)\\ &\leqslant\frac{ c \left( p^{\eta} n \right)^{1/\eta} }{n^{\beta \eta'}} \bigg( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg)^{1/\eta'} \notag\\ &\leqslant \frac{c}{n^{\beta/2} } \bigg( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg), \end{align}\] where \(\mathcal{F}_{\ell}(a)\) is defined by 125 , and in the last inequality we take \(\eta' > 1\) to be sufficiently close to \(1\). This concludes the proof of 170 .

We next prove 171 . By Hölder’s inequality, with \(\eta >1\) and \(\frac{1}{\eta} +\frac{1}{\eta'}=1\), \[\begin{align} \label{app-lem-second-ine001} \mathbb{E}_a \left( |\tilde{f}(\xi_{\tilde{\nu}_{n,t}})|; \; \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t}, \tilde{\nu}_{n,t} \leqslant n \right) \leqslant\mathbb{E}_a^{1/\eta} \left( |\tilde{f}(\xi_{\tilde{\nu}_{n,t}})|^{\eta}; \tilde{\nu}_{n,t} \leqslant n \right) \mathbb{P}_a^{1/\eta'} \left( \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t} \right). \end{align}\tag{173}\] In view of 165 , we write \[\begin{align} \mathbb{E}_a \left( |\tilde{f}(\xi_{\tilde{\nu}_{n,t}})|^\eta; \tilde{\nu}_{n,t} \leqslant n \right) = \sum_{\ell =p+1}^{n} \mathbb{E}_a \left( |\tilde{f}(\xi_{\ell})|^\eta; \tilde{\nu}_{n,t} = \ell \right) \leqslant\sum_{\ell =p+1}^{n} \mathcal{F}_{\ell}(a), \end{align}\] where \(\mathcal{F}_{\ell}(a)\) is defined by 125 . Then, by 173 and Lemma 24, \[\begin{align} & \mathbb{E}_a \left( |\tilde{f}(\xi_{\tilde{\nu}_{n,t}})|; \; \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t}, \tilde{\nu}_{n,t} \leqslant n \right) \notag\\ &\leqslant\bigg( \sum_{\ell =p+1}^{n} \mathcal{F}_{\ell}(a) \bigg)^{1/\eta} \mathbb{P}_a^{1/\eta'} \left( \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t} \right) \notag\\ &\leqslant\bigg( \sum_{\ell =p+1}^{n} \mathcal{F}_{\ell}(a) \bigg)^{1/\eta} \frac{c}{n^{\beta \eta'}} \bigg( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg)^{1/\eta'}. \end{align}\] By taking \(\eta'\) to be sufficiently close to \(1\), we get \[\begin{align} & \mathbb{E}_a \bigg( |\tilde{f}(\xi_{\tilde{\nu}_{n,t}})|; \; \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t}, \tilde{\nu}_{n,t} \leqslant n \bigg) \notag\\ & \leqslant \frac{c}{n^{1+\beta/2} } \left( 1 + \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| \right) \sum_{k = p+1}^n \mathcal{F}_k(a). \end{align}\] This ends the proof of the second inequality 171 of the lemma. ◻

Proof. By Hölder’s inequality, with \(\eta,\eta' >1\) and \(\frac{1}{\eta} +\frac{1}{\eta'}=1\), we have \[\begin{align} \label{CSH-LLL-001} I \leqslant \mathbb{E}_a^{1/\eta} \Bigg( \Bigg| \sum_{j=\tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_j) \Bigg|^{\eta} + |\tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} })|^\eta; p+1< \tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}] \Bigg) \mathbb{P}_a^{1/\eta'} \left( \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t} \right). \end{align}\tag{174}\] Applying 132 and the moment assumption 71 , we get \[\begin{align} \mathbb{E}_a \Bigg( \Bigg| \sum_{j = \tilde{\tau}^{\mathfrak f}_{t}-p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_j) \Bigg|^\eta; \;p+1< \tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}] \Bigg) & = \sum_{\ell =p+2}^{ [n^{1-\varepsilon}] } \mathbb{E}_a \Bigg( \Bigg| \sum_{j = \ell-p+1}^{\ell} h_p(\xi_j) \Bigg|^\eta; \tilde{\tau}^{\mathfrak f}_{t}=\ell \Bigg)\\ & \leqslant\sum_{\ell =p+2}^{ [n^{1-\varepsilon}] } \mathbb{E}_a \Bigg( \Bigg| \sum_{j = \ell-p+1}^{\ell} h_p(\xi_j) \Bigg|^\eta \Bigg) \leqslant c n p^{\eta}. \end{align}\] In view of 125 , it holds that \[\begin{align} \mathbb{E}_a \left( |\tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} })|^\eta; \;p +1 < \tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}] \right) = \sum_{\ell =p+2}^{ [n^{1-\varepsilon}] } \mathbb{E}_a \Bigg( |\tilde{f}(\xi_{ \ell })|^\eta; \tilde{\tau}^{\mathfrak f}_{t}=\ell \Bigg) \leqslant c \sum_{\ell =p+1}^{n} \mathcal{F}_{\ell}(a). \end{align}\] The last two bounds give \[\begin{align} & \mathbb{E}_a^{1/\eta} \Bigg( \Bigg| \sum_{j=\tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_j) \Bigg|^{\eta} + |\tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} })|^\eta; \;p +1 < \tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}] \Bigg) \notag\\ &\leqslant c \bigg( n p^{\eta} + \sum_{\ell =p+1}^{n} \mathcal{F}_{\ell}(a) \bigg)^{1/\eta} \leqslant c n^{1/\eta} \bigg( p + \frac{1}{n} \sum_{\ell =p+1}^{n} \mathcal{F}_{\ell}(a) \bigg)^{1/\eta}. \end{align}\] Using Lemma 24 and taking \(\eta'\) to be sufficiently close to \(1\), we get \[\begin{align} \mathbb{P}_a^{1/\eta'} \left( \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t} \right) \leqslant\frac{c}{n^{\beta/2 }} \bigg( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg)^{1/\eta'} . \end{align}\] Therefore, by 174 and taking into account that \(\eta > 1\) is sufficiently large, we obtain that for \(p\leqslant n^{\varepsilon}\), \[\begin{align} I &\leqslant c n^{1/\eta} \bigg( p + \frac{1}{n} \sum_{\ell =p+1}^{n} \mathcal{F}_{\ell}(a) \bigg)^{1/\eta} \\ &\quad \times \frac{1}{n^{\beta/2 }} \bigg( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg)^{1/\eta'} \\ & \leqslant\frac{c}{n^{1 + \beta/4 }} \left( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| + 1 \right) \sum_{k = p+1}^n \mathcal{F}_k(a), \end{align}\] which concludes the proof of the lemma. ◻

Proof. Using Lemma 16, we get \[\begin{align} & \mathbb{E}_a \left( \left\vert t + \sigma(g_{p} \cdots g_{1}, x) + \sum_{i=1}^n h_p(\xi_i) \right\vert; \tilde{\tau}^{\mathfrak f}_{t} >n, \tilde{\nu}_{n,t} >n^{1-\varepsilon}\right) \notag\\ &\leqslant\mathbb{E}_a \left( t + \sigma(g_{n} \cdots g_{1}, x) + \tilde{f}(\xi_n) - \tilde{f}(\xi_0); \tilde{\tau}^{\mathfrak f}_{t} >n, \tilde{\nu}_{n,t} >n^{1-\varepsilon}\right) \notag\\ &\quad + c \frac{ \max \{t,0\} + \Big|\sigma(g_p\cdots g_1,x) \Big| + |\tilde{f}(a)|}{n^{\beta}} \left( \mathcal{F}_n(a) + |\tilde{f}(a)| \right) + |\tilde{f}(a)| \frac{ \sum_{k = p+1}^n \mathcal{F}_k(a) }{n^3}. \end{align}\] Then, by the Cauchy-Schwarz inequality, we get that, for any \(t \in \mathbb{R},\) \[\begin{align} & \mathbb{E}_a \left( t + \sigma(g_{n} \cdots g_{1}, x) + \tilde{f}(\xi_n) - \tilde{f}(\xi_0); \tilde{\tau}^{\mathfrak f}_{t} >n, \tilde{\nu}_{n,t} >n^{1-\varepsilon}\right) \notag\\ & \leqslant\mathbb{E}_a^{1/2} \left[ \Big( t + \sigma(g_{n} \cdots g_{1}, x) + \tilde{f}(\xi_n) - \tilde{f}(\xi_0) \Big) ^{2}; \tilde{\tau}^{\mathfrak f}_{t} >n \right] \mathbb{P}_a^{1/2}\left( \tilde{\nu}_{n,t}>n^{1-\varepsilon}\right). \end{align}\]By Lemma 10, we have \[\begin{align} & \mathbb{E}_a \left( t + \sigma(g_{n} \cdots g_{1}, x) + \tilde{f}(\xi_n) - \tilde{f}(\xi_0); \tilde{\tau}^{\mathfrak f}_{t} >n, \tilde{\nu}_{n,t} >n^{1-\varepsilon}\right) \notag\\ & \leqslant\mathbb{E}_a^{1/2}\left( \Big( t + \sigma(g_{n} \cdots g_{1}, x) + \tilde{f}(\xi_n) - \tilde{f}(\xi_0) \Big) ^{2}; \tilde{\tau}^{\mathfrak f}_{t} >n \right) \mathbb{P}_a^{1/2}\left( \tilde{\nu}_{n,t}>n^{1-\varepsilon}\right) \notag\\ & \leqslant\left[ \max \{t,0\} + \mathbb{E}_a^{1/2}\left( \Big( \sigma(g_{n} \cdots g_{1}, x) + \tilde{f}(\xi_n) - \tilde{f}(\xi_0) \Big) ^{2}\right) \right] \mathbb{P}_a^{1/2}\left( \tilde{\nu}_{n,t}>n^{1-\varepsilon}\right) \notag \\ & \leqslant c \left( \max \{t,0\} + n + \sigma(g_{p} \cdots g_{1}, x) + \mathcal{F}_n(a) + | \tilde{f}(a) | \right) e^{- \beta n^{\varepsilon}}, \end{align}\] where in the last bound we used Minkowski’s inequality and Lemma 22. ◻

Proof. We shall first prove that there exist constants \(c, c', \varepsilon_0 > 0\) such that for any \(n\geqslant 1\), \(\varepsilon\in \left( 0,\varepsilon_{0}\right)\), \(1\leqslant p\leqslant n^{\varepsilon_0}\), \(t\geqslant n^{1/2-\varepsilon}\) and \(a=(g_0,\ldots, g_{p}, x, q) \in \mathbb{A}_p\), \[\begin{align} \label{Un-xy-Bound-001} W^{\mathfrak f}_{n} (a, t) &\leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) t + |\sigma(g_{p} \cdots g_{1}, x)| \notag \\ & \quad + c e^{- c' n^{1/3} } \Big(1+ t + |\sigma(g_{p} \cdots g_{1}, x)| \Big) \sum_{k = p+1}^n \mathcal{F}_k(a). \end{align}\tag{177}\] Let \(a \in \mathbb{A}_p\) and \(t' = t + \sigma(g_p \cdots g_1, x)\). By 134 and Lemma 14, the sequence of functions \[\begin{align} M_0: = 0, \quad M_n : \xi \mapsto \sum_{i=1}^n h_p(\xi_i), \quad n \geqslant 1 \end{align}\] is a martingale with respect to the probability measure \(\mathbb{P}_a\) and the natural sequence of \(\sigma\)-algebras \((\mathscr G_n)_{n \geqslant 0}\). By 175 and the optional stopping theorem, we get \[\begin{align} \label{bound32G32000} W^{\mathfrak f}_{n} (a, t) &=\mathbb{E}_a \left(t'+M_{n}\right) -\mathbb{E}_a \left( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant n\right) \notag \\ &=t'-\mathbb{E}_a \left( t'+M_{\tilde{\tau}^{\mathfrak f}_{t}};\; \tilde{\tau}^{\mathfrak f}_{t} \leqslant n\right) \notag \\ & = t' - \mathbb{E}_a \left( t'+M_{\tilde{\tau}^{\mathfrak f}_{t}};\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant p \right) - \mathbb{E}_a \left( t'+M_{\tilde{\tau}^{\mathfrak f}_{t}};\;p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n \right) \notag\\ & \leqslant t' - \mathbb{E}_a \left( t'+M_{p};\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant p \right) + \mathbb{E}_a \left( \left\vert t' + M_{\tilde{\tau}^{\mathfrak f}_{t}} \right\vert ;\;p < \tilde{\tau}^{\mathfrak f}_{t} \leqslant n \right). \end{align}\tag{178}\] For the first expectation on the right-hand side of 178 , we have \[\begin{align} \label{Second-Expect} t' - \mathbb{E}_a \left( t'+M_{p};\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant p \right) & = \mathbb{E}_a \left( t'+M_{p};\;\tilde{\tau}^{\mathfrak f}_{t} > p \right) \notag\\ & = t' \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} > p \right) + \mathbb{E}_a \left( M_{p};\;\tilde{\tau}^{\mathfrak f}_{t} > p \right) \notag\\ & \leqslant\max\{t', 0\} + c p \leqslant\left| t + \sigma(g_{p} \cdots g_{1}, x) \right| + cp. \end{align}\tag{179}\] For the second expectation on the right-hand side of 178 , by Lemma 17, there exist constants \(c, c', \varepsilon_0 >0\) such that for any \(\varepsilon\in (0,\varepsilon_0)\), \(a \in \mathbb{A}_p\) and \(t\geqslant n^{1/2-\varepsilon},\) \[\begin{align} \label{First-Expect} \mathbb{E}_a \left( \left\vert t' + M_{\tilde{\tau}^{\mathfrak f}_{t}} \right\vert ;\;p < \tilde{\tau}^f_{t} \leqslant n \right) \leqslant\frac{t}{n^{\varepsilon}} + c e^{- c' n^{1/3} } \Big(1+ \Big| t + \sigma(g_p\cdots g_1,x) \Big|\Big) \sum_{k = p+1}^n \mathcal{F}_k(a), \end{align}\tag{180}\] where we have used the fact that \(p\leqslant n^{\varepsilon_0}\). By substituting 179 and 180 into 178 , we obtain the desired inequality 177 .

Now we proceed to 176 . To do this, we will apply the Markov property and the bound 177 . First, we note that, by 175 , for any \(a\in \mathbb{A}_p\), \(t \in \mathbb{R}\), \(n \geqslant 1\) and \(\varepsilon>0,\) \[\begin{align} \label{bound32J143J2} W^{\mathfrak f}_{n} (a, t) = \mathbb{E}_a \Big( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} >n, \tilde{\nu}_{n,t} > n^{1-\varepsilon} \Big) + \mathbb{E}_a \Big( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} >n, \tilde{\nu}_{n,t} \leqslant n^{1-\varepsilon} \Big), \end{align}\tag{181}\] where \(\tilde{\nu}_{n,t}\) is defined by 165 . For the first term, by Lemma 27, for any \(\varepsilon\in \left( 0,\varepsilon_{0}\right)\), \[\begin{align} \label{bound-J2-ay} & \mathbb{E}_a \left( |t'+M_{n}|;\;\tilde{\tau}^{\mathfrak f}_{t} >n, \tilde{\nu}_{n,t} > n^{1-\varepsilon}\right) \notag\\ &\leqslant c \frac{ \max \{t,0\} + \left| \sigma(g_p\cdots g_1,x) \right| + |\tilde{f}(a)|}{n^{\beta}} \left( \mathcal{F}_n(a) + |\tilde{f}(a)| \right) + |\tilde{f}(a)| \frac{ \sum_{k = p+1}^n \mathcal{F}_k(a) }{n^3}. \end{align}\tag{182}\] For the second term on the right-hand side of 181 , we decompose it into two parts: \[\begin{align} \label{bound32J132001} & \mathbb{E}_a \left( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} > n,\;\tilde{\nu}_{n,t} \leqslant n^{1-\varepsilon} \right) \notag\\ & = \mathbb{E}_a \left( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} > n,\;\tilde{\nu}_{n,t} \leqslant n^{1-\varepsilon}, |\tilde{f}(\xi_{\tilde{\nu}_{n,t}} )| > n^{1/2-\varepsilon} \right) \notag\\ & \quad + \mathbb{E}_a \left( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} > n,\;\tilde{\nu}_{n,t} \leqslant n^{1-\varepsilon}, |\tilde{f}(\xi_{\tilde{\nu}_{n,t}} )| \leqslant n^{1/2-\varepsilon} \right) \notag\\ & =: I_1 (a, t) + I_2 (a, t). \end{align}\tag{183}\] For the first term \(I_1 (a, t)\), by the Cauchy-Schwarz inequality, 125 and Markov’s inequality, we have \[\begin{align} \label{bound32J1232001} I_1 (a, t) & = \sum_{k=p+1}^{ [n^{1-\varepsilon}] } \mathbb{E}_a \left( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} > n,\;\tilde{\nu}_{n,t} = k, |\tilde{f}(\xi_{k} )| > n^{1/2-\varepsilon} \right) \notag\\ & \leqslant\sum_{k=p+1}^{ [n^{1-\varepsilon}] } \mathbb{E}_a^{1/2} \left( |t'+M_{n}|^2 \right) \mathbb{P}_a^{1/2} \left( |\tilde{f}(\xi_{k} )| > n^{1/2-\varepsilon} \right) \notag\\ & \leqslant c' \left( \max\{t, 0\} + |\sigma(g_{p} \cdots g_{1}, x)| + \sqrt{n} \right) e^{-\frac{\alpha}{2} n^{1/2-\varepsilon}} \bigg( \sum_{k=p+1}^{ n } \mathcal{F}_k(a) \bigg)^{1/2} \notag\\ & \leqslant c' \left( 1+\max\{t, 0\} + |\sigma(g_{p} \cdots g_{1}, x)| \right) e^{-c n^{1/2-\varepsilon}} \bigg( \sum_{k=p+1}^{ n } \mathcal{F}_k(a) \bigg)^{1/2} . \end{align}\tag{184}\] For the second term \(I_2 (a, t)\) in 183 , we note that, by 134 and the fact that \(t' = t + \sigma(g_p \cdots g_1, x) = t + \sum_{i=1}^p \sigma_p (\xi_i)\), it holds that \(t' + M_n = t + \sum_{i=1}^{n+p} \sigma_p(\xi_i)\) and \[\begin{align} \label{equality-hp-sigmap} t' + \sum_{i=1}^{k-p} h_p(\xi_i) = t + \sum_{i=1}^{k} \sigma_p(\xi_i). \end{align}\tag{185}\] Hence, applying the Markov property (see Lemma 13), we get \[\begin{align} \label{bound32J132002} I_2 (a, t) & = \sum_{k=p+1}^{ [n^{1-\varepsilon}] } \mathbb{E}_a \left( t' + M_n; \;\tilde{\tau}^{\mathfrak f}_{t} > n,\;\tilde{\nu}_{n,t}=k, |\tilde{f}(\xi_k )| \leqslant n^{1/2-\varepsilon} \right) \notag\\ & = \sum_{k=p+1}^{ [n^{1-\varepsilon}] } \mathbb{E}_a \bigg[ W^{\mathfrak f}_{n-k} \bigg( \xi_k, t' + \sum_{i=1}^{k-p} h_p(\xi_i) + \tilde{f}(\xi_k)- \tilde{f}(\xi_0) \bigg); A_{n,k} \bigg], \end{align}\tag{186}\] where \[\begin{align} \label{def-Ank} A_{n,k} = \left\{ \tilde{\tau}^{\mathfrak f}_{t} >k, \;\tilde{\nu}_{n,t}=k, |\tilde{f}(\xi_k )| \leqslant n^{1/2-\varepsilon} \right\}. \end{align}\tag{187}\] Using the definition of the stopping times \(\tilde{\nu}_{n, t}\) and \(\tilde{\tau}^{\mathfrak f}_{t}\) (see 165 and 126 ) and 185 , one can verify that on the event \(A_{n, k}\), we have simultaneously \(| t' + \sum_{i=1}^{k-p} h_p(\xi_i) - \tilde{f}(\xi_0) | \geqslant 2 n^{1/2-\varepsilon}\), \(t' + \sum_{i=1}^{k-p} h_p(\xi_i) + \tilde{f}(\xi_k ) - \tilde{f}(\xi_0)\geqslant 0\) and \(|\tilde{f}(\xi_k )| \leqslant n^{1/2-\varepsilon}\). All these bounds imply that \(t' + \sum_{i=1}^{k-p} h_p(\xi_i) + \tilde{f}(\xi_k)- \tilde{f}(\xi_0) \geqslant n^{1/2-\varepsilon}\). Thus, we have the inclusion: \[\begin{align} A_{n,k } \subseteq \bigg\{ t' + \sum_{i=1}^{k-p} h_p(\xi_i) + \tilde{f}(\xi_k)- \tilde{f}(\xi_0) \geqslant n^{1/2-\varepsilon}\bigg\}. \end{align}\] On the event \(A_{n,k}\), applying 177 , there exist constants \(c, c', \varepsilon_0 > 0\) such that for any \(n\geqslant 1\), \(\varepsilon\in \left( 0,\varepsilon_{0}\right)\), \(1\leqslant p\leqslant n^{\varepsilon_0}\), \(k \in [p + 1, [n^{1-\varepsilon}]]\), \(t\geqslant n^{1/2-\varepsilon}\) and \(a \in \mathbb{A}_p\), \[\begin{align} & W^{\mathfrak f}_{n-k} \bigg( \xi_k, t' + \sum_{i=1}^{k-p} h_p(\xi_i) + \tilde{f}(\xi_k)- \tilde{f}(\xi_0) \bigg) \notag\\ &\leqslant\bigg( 1+ \frac{c}{(n-k)^{\varepsilon}} \bigg) \bigg(t' + \sum_{i=1}^{k-p} h_p(\xi_i)+ \tilde{f}(\xi_k)- \tilde{f}(\xi_0) \bigg) + |\sigma(g_{p} \cdots g_{1}, x)| \notag\\ & \quad + c e^{- c' (n-k)^{1/3} } \bigg(1+t' + \sum_{i=1}^{k-p} h_p(\xi_i)+ \tilde{f}(\xi_k)- \tilde{f}(\xi_0)+ |\sigma(g_{p} \cdots g_{1}, x)| \bigg) \sum_{j = p+1}^{n} \mathcal{F}_j(a) \notag\\ &\leqslant\bigg( 1+ \frac{c}{n^{\varepsilon}} \bigg) \bigg(t' + \sum_{i=1}^{k-p} h_p(\xi_i)+ \tilde{f}(\xi_k)- \tilde{f}(\xi_0) \bigg) + |\sigma(g_{p} \cdots g_{1}, x)| \notag\\ & \quad + c e^{- c' n^{1/4} } \bigg(1+t' + \sum_{i=1}^{k-p} h_p(\xi_i)+ \tilde{f}(\xi_k)- \tilde{f}(\xi_0)+ |\sigma(g_{p} \cdots g_{1}, x)| \bigg) \sum_{j = p+1}^{n} \mathcal{F}_j(a), \end{align}\] which, together with the definition of \(A_{n, k}\) (cf.@eq:def-Ank ), implies that \[\begin{align} \label{bound32E001} & \mathbb{E}_a \bigg[ W^{\mathfrak f}_{n-k} \bigg( \xi_k, t' + \sum_{i=1}^{k-p} h_p(\xi_i) + \tilde{f}(\xi_k)- \tilde{f}(\xi_0) \bigg); A_{n,k} \bigg] \notag\\ & \leqslant\bigg( 1+ \frac{c}{n^{\varepsilon}} \bigg) \mathbb{E}_a \bigg( t' + \sum_{i=1}^{k-p} h_p(\xi_i)- \tilde{f}(\xi_0); \; A_{n,k} \bigg) + c \mathbb{E}_a \left( |\tilde{f}(\xi_k)|; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k \right) \notag \\ & \quad + |\sigma(g_{p} \cdots g_{1}, x)| \mathbb{P}_a \Big( \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k \Big) \notag \\ & \quad + c e^{- c' n^{1/4} } \bigg( \sum_{j = p+1}^{n} \mathcal{F}_j(a) \bigg) \notag\\ & \qquad \times \mathbb{E}_a \bigg( 1 + t' + \sum_{i=1}^{k-p} | h_p(\xi_i) | +|\tilde{f}(\xi_0)|+|\tilde{f}(\xi_k)| + |\sigma(g_{p} \cdots g_{1}, x)|; \; A_{n,k} \bigg). \end{align}\tag{188}\] Substituting 188 into 186 , we obtain \[\begin{align} \label{inequality-J11-hh} I_2 (a, t) &\leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) \sum_{k=p+1}^{ [ n^{1-\varepsilon}] } \mathbb{E}_a \bigg( t' + \sum_{i=1}^{k-p} h_p(\xi_i)- \tilde{f}(\xi_0); \; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k, |\tilde{f}(\xi_k )| \leqslant n^{1/2-\varepsilon} \bigg) \notag \\ & \quad + c \mathbb{E}_a \Big(|f(\xi_{\tilde{\nu}_{n,t}})|; \tilde{\tau}^{\mathfrak f}_{t} >\tilde{\nu}_{n,t}, \tilde{\nu}_{n,t} \leqslant n^{1-\varepsilon} \Big) + |\sigma(g_{p} \cdots g_{1}, x)| \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} > \tilde{\nu}_{n,t} \right) \notag \\ &\quad + c e^{- c' n^{1/4} } \bigg( \sum_{j = p+1}^{n} \mathcal{F}_j(a) \bigg) \notag\\ & \qquad \times \sum_{k=p+1}^{ [ n^{1-\varepsilon}] } \mathbb{E}_a \bigg( 1+ t' + \sum_{i=1}^{k - p} | h_p(\xi_i) | + |\tilde{f}(\xi_0)|+|\tilde{f}(\xi_k)| + |\sigma(g_{p} \cdots g_{1}, x)|; \; A_{n,k} \bigg). \end{align}\tag{189}\] For the second term on the right-hand side of 189 , by the second inequality 171 of Lemma 25, there exist constants \(c, \beta, \varepsilon> 0\) such that, for any \(a \in \mathbb{A}_p\), \(t \in \mathbb{R}\), \(n\geqslant 1\) and \(p\leqslant n^{\varepsilon}\), \[\begin{align} \label{xi95nu32bound32001} &\mathbb{E}_a \left( |\tilde{f}(\xi_{\tilde{\nu}_{n,t}})|; \; \tilde{\tau}^{\mathfrak f}_{t} > \tilde{\nu}_{n,t}, \tilde{\nu}_{n,t} \leqslant n \right) \notag\\ &\leqslant c \left( 1 + \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| +|\tilde{f}(a)| \right) \frac{1}{n^{1 + \beta }} \sum_{j = p+1}^n \mathcal{F}_j(a). \end{align}\tag{190}\] For the third term on the right-hand side of 189 , by Lemma 24, we get that for any \(1\leqslant p\leqslant n^{\varepsilon}\) and \(t \in \mathbb{R}\), \[\begin{align} \label{SecondinJ11-001} \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} > \tilde{\nu}_{n,t} \right) \leqslant \frac{c}{n^{\beta }} \bigg( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg). \end{align}\tag{191}\] For the last term on the right-hand side of 189 , using the definition of \(A_{n, k}\) (cf.@eq:def-Ank ), the fact that \(t' = t + \sigma(g_p \cdots g_1, x)\) and the moment assumption 71 , we obtain \[\begin{align} \label{ThirdinJ11-001} & \sum_{k=p+1}^{ [ n^{1-\varepsilon}] } \mathbb{E}_a \bigg( 1+ t' + \sum_{i=1}^{k - p} | h_p(\xi_i) | + |\tilde{f}(\xi_0)|+|\tilde{f}(\xi_k)| + |\sigma(g_{p} \cdots g_{1}, x)|; \; A_{n,k} \bigg) \notag\\ & \leqslant\sum_{k=p+1}^{ [ n^{1-\varepsilon}] } \mathbb{E}_a \bigg( 1+\max \{t,0\} + \sum_{i=1}^{k - p} | h_p(\xi_i) |+|\tilde{f}(\xi_0)|+|\tilde{f}(\xi_k)| + |\sigma(g_{p} \cdots g_{1}, x)|; \notag\\ & \qquad\qquad\qquad \tilde{\tau}^{\mathfrak f}_{t} > k , \tilde{\nu}_{n,t}=k \bigg) \notag \\ & \leqslant \mathbb{E}_a \bigg( 1+ \max \{t,0\} + \sum_{i=1}^{\tilde{\nu}_{n,t}-p} | h_p(\xi_i) |+|\tilde{f}(\xi_0)| + |\tilde{f}(\xi_{\tilde{\nu}_{n,t}})| + |\sigma(g_{p} \cdots g_{1}, x)|; \; \tilde{\nu}_{n,t} \leqslant n^{1-\varepsilon} \bigg) \notag \\ & \leqslant\bigg( 1 + \max \{t,0\} + cn + |\tilde{f}(a)| + c \sum_{j = p+1}^n \mathcal{F}_j(a) + |\sigma(g_{p} \cdots g_{1}, x)| \bigg) \notag\\ & \leqslant\bigg( \max \{t,0\} + |\tilde{f}(a)| + |\sigma(g_{p} \cdots g_{1}, x)| + c \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg), \end{align}\tag{192}\] where in the last inequality we used the fact that \(p \in [1, n^{\varepsilon}]\) and \(\mathcal{F}_j(a) > 1\) for any \(j \geqslant 1\). Substituting 190 , 191 and 192 into 189 , we obtain \[\begin{align} \label{inequality-J11-hh-002} I_2 (a, t) &\leqslant\left( 1+ \frac{c}{n^{\varepsilon}} \right) \sum_{k=p+1}^{ [ n^{1-\varepsilon}] } \mathbb{E}_a \bigg( t' + \sum_{i=1}^{k-p} h_p(\xi_i) - \tilde{f}(\xi_0); \; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k, |\tilde{f}(\xi_k )| \leqslant n^{1/2-\varepsilon} \bigg) \notag \\ & \quad + \frac{c}{n^{1+\beta}} \left( 1 + |\sigma(g_p \cdots g_1, x)| \right) \bigg(\sum_{j = p+1}^n \mathcal{F}_j(a) \bigg) \notag \\ & \qquad \times \bigg( |\tilde{f}(a)|+\max \{t,0\} + |\sigma(g_{p} \cdots g_{1}, x)| + \frac{1}{n}\sum_{j = p+1}^n \mathcal{F}_j(a) \bigg). \end{align}\tag{193}\] Now we proceed by completing the sum \(t' + \sum_{i=1}^{k-p} h_p(\xi_i) - \tilde{f}(\xi_0)\) in the first term on the right-hand side of 193 to obtain the full martingale expression \(t' + \sum_{i=1}^{k} h_p(\xi_i) - \tilde{f}(\xi_0)\). To achieve this, we will address the difference between the two sums, which is exactly \(\sum_{i=k-p+1}^{k} h_p(\xi_i).\) Additionally, we want to transition from the event \(\{|\tilde{f}(\xi_k )| \leqslant n^{1/2-\varepsilon}\}\) to its complement, thus introducing an extra term. This leads us to decompose the first term on the right-hand side of 193 as follows: \[\begin{align} \label{decom-J11-123} & \mathbb{E}_a \bigg( t' + \sum_{i=1}^{k-p} h_p(\xi_i) - \tilde{f}(\xi_0); \; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k, |\tilde{f}(\xi_k )| \leqslant n^{1/2-\varepsilon} \bigg) \notag \\ & = \mathbb{E}_a \bigg( t' + \sum_{i=1}^{k} h_p(\xi_i) - \tilde{f}(\xi_0); \; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k \bigg) \notag \\ & \quad - \mathbb{E}_a \bigg( \sum_{i=k-p+1}^{k} h_p(\xi_i); \; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k \bigg) \notag \\ & \quad - \mathbb{E}_a \bigg( t' + \sum_{i=1}^{k-p} h_p(\xi_i) - \tilde{f}(\xi_0); \; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k, |\tilde{f}(\xi_k )| > n^{1/2-\varepsilon} \bigg) \notag \\ & =: J_1(k) - J_2(k) - J_3(k). \end{align}\tag{194}\] The first term \(J_1(k)\) contributes directly to one part of the desired result. For the second term \(J_2(k)\), by the first inequality 170 of Lemma 25, we have \[\begin{align} \label{bound-J-112-001} \sum_{k= p+1}^{ [ n^{1-\varepsilon}] } |J_2(k)| \leqslant\frac{c}{n^{\beta }} \left( \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f} (a)| + \frac{1}{n}\sum_{j = p+1}^n \mathcal{F}_j(a) \right). \end{align}\tag{195}\] For the third term \(J_3(k)\), in view of 126 and 185 , it holds that \(\tilde{\tau}^{\mathfrak f}_{t} >k\) implies \(t' + \sum_{i=1}^{k-p} h_p(\xi_i) + \tilde{f}(\xi_k ) - \tilde{f}(\xi_0) \geqslant 0\), so that \[\begin{align} - J_3(k) & \leqslant\mathbb{E}_a \left( \tilde{f}(\xi_k); \; \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k, |\tilde{f}(\xi_k )| > n^{1/2-\varepsilon} \right) \notag\\ & \leqslant\mathbb{E}_a \left( |\tilde{f}(\xi_k)|; \; |\tilde{f}(\xi_k )| > n^{1/2-\varepsilon} \right) \notag\\ & \leqslant c e^{ - \frac{\alpha}{2} n^{1/2-\varepsilon} } \mathcal{F}_k(a). \end{align}\] Hence \[\begin{align} \label{bound-J-113-001} \sum_{k=p+1}^{ [ n^{1-\varepsilon}] } - J_3(k) \leqslant c e^{ - \frac{\alpha}{2} n^{1/2-\varepsilon} } \sum_{j = p+1}^n \mathcal{F}_j(a) \leqslant\frac{c}{n^{1+\beta}} \sum_{j = p+1}^n \mathcal{F}_j(a). \end{align}\tag{196}\] Since \(\mathcal{F}_j(a) > 1\) and \(1 \leqslant p \leqslant n^{\varepsilon}\), it holds that \(\sum_{j = p+1}^n \mathcal{F}_j(a) \geqslant\frac{n}{2}\). Therefore, combining 193 , 194 , 195 and 196 , and noting that \(M_k= \sum_{i=1}^{k} h_p(\xi_i)\) (by 134 ), we obtain \[\begin{align} \label{Start32induction-001} I_2 (a, t) & \leqslant\left( 1+\frac{c}{n^{\varepsilon}}\right) \sum_{k=p+1}^{ [ n^{1-\varepsilon} ] } \mathbb{E}_a \left( t' + M_k - \tilde{f}(\xi_0); \tilde{\tau}^{\mathfrak f}_{t} >k, \tilde{\nu}_{n,t}=k \right) \notag\\ & \quad + \frac{c}{n^{1+\beta}} ( 1 + |\sigma(g_{p} \cdots g_{1}, x)| ) \bigg( \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg) \notag \\ & \quad\quad \times \bigg( |\tilde{f}(a)| + \max \{t,0\} + |\sigma(g_{p} \cdots g_{1}, x)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg). \end{align}\tag{197}\] Now we deal with the first expectation on the right-hand side of 197 . We shall prove that for \(k\in [p+1, [n^{1-\varepsilon}] ]\), \[\begin{align} \label{Equ-desired} & \mathbb{E}_a\left( t' + M_{k} - \tilde{f}(\xi_0);\; \tilde{\tau}^{\mathfrak f}_{t} > k, \tilde{\nu}_{n,t} = k \right) \notag\\ & \leqslant\mathbb{E}_a\left( t'+M_{ [n^{1-\varepsilon}]} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}], \tilde{\nu}_{n,t}=k\right) \notag\\ & \quad + \mathbb{E}_a \Bigg( - \sum_{i = \tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) - \tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} }); \; k + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}], \tilde{\nu}_{n,t} = k \Bigg). \end{align}\tag{198}\] Indeed, since the event \(\{ \tilde{\nu}_{n,t} = k \}\) is in \(\mathscr G_{k}\), we have \(\mathbb{E}_a (M_{ [n^{1-\varepsilon}]}; \tilde{\nu}_{n,t} = k) = \mathbb{E}_a (M_{ k}; \tilde{\nu}_{n,t} = k)\), so that \[\begin{align} \label{intermediate32ident-001} & \mathbb{E}_a \left( t' + M_{ [n^{1-\varepsilon}]} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}], \tilde{\nu}_{n,t}=k\right) - \mathbb{E}_a \left( t'+M_{k} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k, \tilde{\nu}_{n,t}=k\right) \notag\\ & = \mathbb{E}_a \left( t'+M_{k}- \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > k, \tilde{\nu}_{n,t}=k\right) \notag\\ & \qquad - \mathbb{E}_a\left( t'+M_{ [n^{1-\varepsilon}]} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}], \tilde{\nu}_{n,t}=k\right). \end{align}\tag{199}\] Using 199 , we see that proving 198 is equivalent to showing the following inequality: \[\begin{align} \label{Equ-00a} & \mathbb{E}_a\left( t'+M_{ [n^{1-\varepsilon}]} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}], \tilde{\nu}_{n,t}=k\right) \notag\\ & \leqslant\mathbb{E}_a\left( t'+M_{k} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k, \tilde{\nu}_{n,t}=k\right) \notag\\ & \quad + \mathbb{E}_a \Bigg( - \sum_{i = \tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) - \tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} });\; k + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}], \tilde{\nu}_{n,t} = k \Bigg). \end{align}\tag{200}\] Now we prove 200 by an induction argument. Let \(j \in [k + 1, [n^{1-\varepsilon}]]\) be an integer. Then \[\begin{align} & \mathbb{E}_a \left( t' + M_j - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant j, \tilde{\nu}_{n,t} = k \right) \notag\\ & = \mathbb{E}_a \left( t' + M_j - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant j -1, \tilde{\nu}_{n,t} = k \right) + \mathbb{E}_a \left( t' + M_j - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} = j, \tilde{\nu}_{n,t} = k \right). \end{align}\] As the event \(\{ \tilde{\tau}^{\mathfrak f}_{t} \leqslant j - 1, \tilde{\nu}_{n,t}=k \}\) is in \(\mathscr G_{j-1}\), by the martingale property, we have \[\begin{align} \mathbb{E}_a \left( t' + M_j - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant j -1, \tilde{\nu}_{n,t} = k \right) = \mathbb{E}_a \left( t' + M_{j -1} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant j -1, \tilde{\nu}_{n,t} = k \right). \end{align}\] Besides, by 126 and 134 , on the set \(\{\tilde{\tau}^{\mathfrak f}_{t} = j \}\), it holds that \(t' + M_{j - p} + \tilde{f}(\xi_j) - \tilde{f}(\xi_0)<0\), which leads to \[\begin{align} \mathbb{E}_a \left( t' + M_j - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} = j, \tilde{\nu}_{n,t}=k\right) \leqslant\mathbb{E}_a \bigg( - \sum_{i = j - p + 1}^j h_p(\xi_i) - \tilde{f}(\xi_j); \;\tilde{\tau}^{\mathfrak f}_{t} = j, \tilde{\nu}_{n,t}=k \bigg). \end{align}\] Hence, for any integer \(j \in [k+1, [n^{1-\varepsilon}]]\), \[\begin{align} & \mathbb{E}_a \left( t' + M_j - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant j, \tilde{\nu}_{n,t} = k \right) \notag\\ & \leqslant\mathbb{E}_a \left( t'+M_{j -1} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant j -1, \tilde{\nu}_{n,t} = k \right) \notag\\ & \quad + \mathbb{E}_a \bigg( - \sum_{i = j - p + 1}^j h_p(\xi_i) - \tilde{f}(\xi_j);\;\tilde{\tau}^{\mathfrak f}_{t} = j, \tilde{\nu}_{n,t} = k \bigg). \end{align}\] Summing these bounds up in \(j \in [k+1, [n^{1-\varepsilon}]]\), we get \[\begin{align} & \mathbb{E}_a \left( t'+M_{ [n^{1-\varepsilon}]} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}], \tilde{\nu}_{n,t} = k \right) \notag\\ & \leqslant\mathbb{E}_a \left( t'+M_{k} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k, \tilde{\nu}_{n,t}=k\right) \notag\\ & \quad + \sum_{j = k+1}^{ [n^{1-\varepsilon}] } \mathbb{E}_a \bigg( - \sum_{i = j-p+1}^h h_p(\xi_i) - \tilde{f}(\xi_j); \;\tilde{\tau}^{\mathfrak f}_{t} = j, \tilde{\nu}_{n,t} = k \bigg) \notag\\ & = \mathbb{E}_a \left( t'+M_{k} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} \leqslant k, \tilde{\nu}_{n,t}=k\right) \notag\\ & \quad + \mathbb{E}_a \Bigg( - \sum_{i = \tilde{\tau}^{\mathfrak f}_{t} - p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) - \tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} }); \; k + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}], \tilde{\nu}_{n,t} = k \Bigg). \end{align}\] This shows 200 and therefore 198 holds.

Summing over \(k\) in 198 , we obtain \[\begin{align} \label{decom-H-H1H2} H (a, t): & = \sum_{k=p+1}^{ [ n^{1-\varepsilon} ] } \mathbb{E}_a\left( t'+M_{k} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > k, \tilde{\nu}_{n,t}=k\right) \notag\\ & \leqslant\mathbb{E}_a\left( t'+M_{ [n^{1-\varepsilon}]} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}], \tilde{\nu}_{n,t} \leqslant[n^{1-\varepsilon}] \right) \notag\\ & \quad + \mathbb{E}_a \Bigg( \Bigg| \sum_{i = \tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \Bigg| + |\tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} })|; \; \tilde{\nu}_{n,t} + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}] \Bigg). \end{align}\tag{201}\] For the first term on the right-hand side of 201 , by the definition of \(W^{\mathfrak f}_{n}(a,t)\) (cf.@eq:EXPECT-E95x-002 ) and the fact that \(\tilde{f}(\xi_0) = \tilde{f}(a)\), we have \[\begin{align} \label{bound-H1-abc} & \mathbb{E}_a \left( t' + M_{ [n^{1-\varepsilon}]} - \tilde{f}(\xi_0);\;\tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}], \tilde{\nu}_{n,t} \leqslant[n^{1-\varepsilon}] \right) \notag \\ & = W^{\mathfrak f}_{[n^{1-\varepsilon}] } (a, t) - \tilde{f}(a) \mathbb{P}_a(\tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}], \tilde{\nu}_{n,t} \leqslant[n^{1-\varepsilon}] ) + H (a, t) \notag \\ & \leqslant W^{\mathfrak f}_{[n^{1-\varepsilon}] } (a, t) + |\tilde{f}(a)| \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}] \right) + K (a, t), \end{align}\tag{202}\] where, for short, we denote \[\begin{align} K (a, t) = - \mathbb{E}_a \left( t' + M_{ [n^{1-\varepsilon}]};\;\tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}], \tilde{\nu}_{n,t} > [n^{1-\varepsilon}] \right). \end{align}\] For the second term on the right-hand side of 202 , by Lemma 15, there exist constants \(c, \beta, \varepsilon> 0\) such that, for any \(a \in \mathbb{A}_p\), \(t \in \mathbb{R}\), \(n\geqslant 1\) and \(p\leqslant n^{\varepsilon}\), \[\begin{align} \label{bound32Q95n32001} \mathbb{P}_a \left( \tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}] \right) \leqslant\frac{c}{n^{\beta}} \bigg( \max \{t,0\} + |\sigma(g_p\cdots g_1,x)| +|\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg). \end{align}\tag{203}\] For the last term \(K (a, t)\) on the right-hand side of 202 , by 126 , 134 and the fact that \(t' = t + \sigma(g_p \cdots g_1, x) = t + \tilde{f}(\xi_0)\), on the set \(\{ \tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}] \}\), we have \(t' + M_{ [n^{1-\varepsilon}] - p } + \tilde{f}(\xi_{[n^{1-\varepsilon}]} ) \geqslant 0\), so that \[\begin{align} K(a, t) \leqslant\mathbb{E}_a \Bigg( \sum_{i = [n^{1-\varepsilon}] - p + 1}^{ [n^{1-\varepsilon}] } |h_p(\xi_i)| + |\tilde{f}(\xi_{[n^{1-\varepsilon}]} )|; \;\tilde{\tau}^{\mathfrak f}_{t} > [n^{1-\varepsilon}], \tilde{\nu}_{n,t} > [n^{1-\varepsilon}] \Bigg). \end{align}\] By the Cauchy-Schwarz inequality, the moment assumption 71 and Lemma 22, it follows that there exist constants \(\beta, \varepsilon, c_{\varepsilon} > 0\) such that, for any \(a \in \mathbb{A}_p\), \(t \in \mathbb{R}\), \(n \geqslant 1\) and \(p \leqslant n^{\varepsilon}\), \[\begin{align} \label{bound32for32R95n32001} K(a, t) & \leqslant\mathbb{E}_a^{1/2} \Bigg( \sum_{i = [n^{1-\varepsilon}] - p + 1}^{ [n^{1-\varepsilon}] } |h_p(\xi_i)| + |\tilde{f}(\xi_{[n^{1-\varepsilon}]} )| \Bigg)^2 \mathbb{P}_a^{1/2}\left( \tilde{\nu}_{n,t} > [n^{1-\varepsilon}] \right) \notag\\ & \leqslant c_{\varepsilon} e^{-\beta n^{\varepsilon}} \mathcal{F}_{[n^{1-\varepsilon}]} (a). \end{align}\tag{204}\] For the last term on the right-hand side of 201 , by Lemma 26, there exist constants \(c, \beta, \varepsilon> 0\) such that, for any \(a \in \mathbb{A}_p\), \(t \in \mathbb{R}\), \(n\geqslant 1\) and \(p\leqslant n^{\varepsilon}\), \[\begin{align} \label{bound32H2-001} & \mathbb{E}_a \Bigg( \Bigg| \sum_{i = \tilde{\tau}^{\mathfrak f}_{t} -p+1}^{\tilde{\tau}^{\mathfrak f}_{t}} h_p(\xi_i) \Bigg| + |\tilde{f}(\xi_{ \tilde{\tau}^{\mathfrak f}_{t} })|; \; \tilde{\nu}_{n,t} + 1 \leqslant\tilde{\tau}^{\mathfrak f}_{t} \leqslant[n^{1-\varepsilon}] \Bigg)\notag \\ & \leqslant\frac{c}{n^{1 + \beta }} \left( 1 + \max \{t,0\} + | \sigma(g_p\cdots g_1, x)| + |\tilde{f}(a)| \right) \sum_{k = p+1}^n \mathcal{F}_k(a). \end{align}\tag{205}\] Substituting 202 , 203 , 204 and 205 into 201 , and taking into account that \(\sum_{j = p+1}^n \mathcal{F}_j(a) \geqslant\frac{n}{2}\) for \(1 \leqslant p \leqslant n^{\varepsilon}\), we get \[\begin{align} & H (a, t) \leqslant W^{\mathfrak f}_{[n^{1-\varepsilon}] } (a, t) \notag\\ & + \frac{c}{n^\beta} \bigg( |\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg) \bigg( \max \{t,0\} + |\sigma(g_p\cdots g_1,x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{k = p+1}^n \mathcal{F}_k(a) \bigg). \end{align}\] Recalling the definition of \(H(a, t)\) (cf.@eq:decom-H-H1H2 ) and substituting the latter bound into 197 gives \[\begin{align} I_2 (a, t) &\leqslant\left( 1+\frac{c_{\varepsilon} }{n^{\varepsilon}}\right) W^{\mathfrak f}_{[n^{1-\varepsilon}] } (a, t) \notag + \frac{c}{n^{\beta}} ( 1 + |\sigma(g_{p} \cdots g_{1}, x)|) \bigg( |\tilde{f}(a)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg) \notag \\ & \qquad\qquad \times \bigg( \max \{t,0\} + |\sigma(g_{p} \cdots g_{1}, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg). \end{align}\] This, together with 183 and 184 , implies that \[\begin{align} & \mathbb{E}_a \left( t'+M_{n};\;\tilde{\tau}^{\mathfrak f}_{t} > n,\;\tilde{\nu}_{n,t} \leqslant n^{1-\varepsilon} \right) \notag\\ & \leqslant\left( 1+\frac{c_{\varepsilon} }{n^{\varepsilon}}\right) W^{\mathfrak f}_{[n^{1-\varepsilon}] } (a, t) + \frac{c}{n^{\beta}} ( 1 + |\sigma(g_{p} \cdots g_{1}, x)|) \bigg( |\tilde{f}(a)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg) \notag \\ & \qquad\qquad \times \bigg( \max \{t,0\} + |\sigma(g_{p} \cdots g_{1}, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg) . \end{align}\] Implementing this and the bound 182 into 181 , we obtain \[\begin{align} W^{\mathfrak f}_{n} (a, t) &\leqslant\left( 1+\frac{c_{\varepsilon} }{n^{\varepsilon}}\right) W^{\mathfrak f}_{[n^{1-\varepsilon}] } (a, t) + \frac{c}{n^{\beta}} ( 1 + |\sigma(g_{p} \cdots g_{1}, x)| ) \bigg( |\tilde{f}(a)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg) \notag \\ & \qquad\qquad \times \bigg( \max \{t,0\} + |\sigma(g_{p} \cdots g_{1}, x)| + |\tilde{f}(a)| + \frac{1}{n} \sum_{j = p+1}^n \mathcal{F}_j(a) \bigg). \end{align}\] This finishes the proof of Lemma 28. ◻

Proof. The assertion of Proposition 20 is obtained from 120 , 127 , Corollary 12 and Lemma 28. Note that \(\mathbb{E} \sigma(g_{p} \cdots g_{1}, x)^2 \leqslant c p\), by the moment assumption 71 and the centering assumption 72 . Since \(a=(g_0,\ldots,g_{p},x,0) \in \mathbb{A}_p\), integrating both sides of 176 in Lemma 28 (with \(\alpha/4\) instead of \(\alpha\)) with respect to the measure \(\mu(dg_0) \ldots \mu(dg_{p}) \nu(dx)\) and using 127 yields that, for any \(t \in \mathbb{R}\), \(n\geqslant 1\) and \(1\leqslant p \leqslant n^{\varepsilon/2}\), \[\begin{align} \label{Expect-E95x001-002} \int_{\mathbb{X}} \mathbb{E} \left( t+S_{n+p}^x ; \tau^{\mathfrak f}_{x, t}>n \right) \nu(dx) & \leqslant \left( 1+\frac{c_{\varepsilon}}{n^{\varepsilon}}\right) \int_{\mathbb{X}} \mathbb{E} \left( t+S_{[n^{1-\varepsilon}]+p}^x ; \tau^{\mathfrak f}_{x, t}> [n^{1-\varepsilon}] \right) \nu(dx) \notag \\ & \quad + \frac{c_{\varepsilon}}{n^{\varepsilon/2}} \left(1+ \max \{t,0\} \right) C_{\alpha}(\mathfrak f), \end{align}\tag{206}\] where we have used the fact that the measure \(\nu\) is \(\mu\)-stationary, as in 139 . Using Lemma 12, we can slightly modify the expectations on both sides of 206 to get \[\begin{align} U_n^{\mathfrak f}(t)&= \int_{\mathbb{X}} \mathbb{E} \left( t+S_{n}^x + f_n( T^n (\omega,x) ) - f_0(\omega,x) ; \tau^{\mathfrak f}_{x, t}>n \right) \nu(dx) \notag \\ & \leqslant \left( 1+\frac{c_{\varepsilon}}{n^{\varepsilon}}\right) U_{[n^{1-\varepsilon}]}^{\mathfrak f}(t) + \frac{c_{\varepsilon}}{n^{\varepsilon/2}} \left(1+ \max \{t,0\} \right) C_{\alpha}(\mathfrak f). \end{align}\] This concludes the proof of Proposition 20. ◻

Proof. Let \(\varepsilon>0\) be as in Proposition 20. Pick \(n\geqslant 1\) and set \(p=[n^{\varepsilon/2}]\). Using the upper bound from Proposition 14 with \(\gamma=\varepsilon\) and \(\delta=\varepsilon/2\), we get \[\begin{align} \label{AA-proof-propapprox-001} U^{\mathfrak f}_n (t) \leqslant U^{\mathfrak f_p}_n (t+ n^{-\varepsilon}) + c \left( \max \{t,0\} + C_{\alpha}(\mathfrak f) \right) e^{ -\beta n^{\varepsilon/2}} D_{\alpha,\beta}(\mathfrak f). \end{align}\tag{207}\] By Proposition 20, we have \[\begin{align} U^{\mathfrak f_p}_{n} (t+ n^{-\varepsilon}) &\leqslant\left( 1+\frac{c }{n^{\varepsilon}}\right) U^{\mathfrak f_p}_{[n^{1-\varepsilon}] } (t+ n^{-\varepsilon}) + c (1+ \max \{t,0\}) \frac{1}{n^{\varepsilon/2 }} C_{\alpha}(\mathfrak f_p). \end{align}\] By Jensen’s inequality, we have \(C_{\alpha}(\mathfrak f_p) \leqslant C_{\alpha}(\mathfrak f)\). Hence, using the lower bound from Proposition 14, we obtain \[\begin{align} U^{\mathfrak f_p}_{n} (t + n^{-\varepsilon}) \leqslant\left( 1+\frac{c }{n^{\varepsilon}}\right) U^{\mathfrak f}_{[n^{1-\varepsilon}] } (t+ 2n^{-\varepsilon}) + c (\max \{t,0\} + C_{\alpha}(\mathfrak f)) \frac{1}{n^{\varepsilon/2 }} D_{\alpha,\beta}(\mathfrak f). \end{align}\] Combining this with 207 , we deduce the assertion of Proposition 21. ◻

Proof. This is a direct consequence of Propositions 16 and 21, by using Lemma 12. Note that the property 110 in Lemma 12 holds due to Lemma 10 and our moment assumption. ◻

Proof. The demonstration is derived using a similar approach to that of Proposition 20. This involves multiplying the inequality from Lemma 28 by \(\theta(g_1, \ldots, g_p)\) and then applying the integral formula 164 . ◻

Proof of Proposition 18. The bound ?? follows from Propositions 23 and 15. ◻

Proof of the second part of Theorem 10. Inequality 86 follows from Propositions 18 and 17, by using Lemmas 10 and 12 as in the proof of Proposition 22. ◻