1 Introduction↩︎

Let \(X_0 = 0\), and for \(n \geq 0\) \[\label{eq:X-def} X_{n+1} = \sum_{i=1}^{X_n} A_{n+1,i} + B_{n+1} =: \theta_{n+1} \circ X_n + B_{n+1},\tag{1}\] where \(\{A_{n,i}: n \geq 1, i \geq 1 \}\) are nonnegative integer-valued iid (independent, identically distributed) random variables, and independently of the \(A\)’s, \(\{ B_n : n \geq 1 \}\) is an iid sequence of nonnegative integer-valued random variables, and \(A\) and \(B\) are the corresponding generic copies. Then \(A_{n+1,i}\) is the number of offspring of individual \(i\) in generation \(n\), and \(B_{n+1}\) is the number of immigrants. Put \[\label{eq:f-g-def} f(s) = \mathbf{E}(s^A), \quad g(s) = \mathbf{E}(s^B), \quad s \in [0,1],\tag{2}\] for the generating function of the offspring and the immigration distribution, respectively. Foster and Williamson [1] proved that \(X_n \stackrel{\mathcal{D}}{\to} X_\infty\) for some finite random variable \(X_\infty\), where \(\stackrel{\mathcal{D}}{\to}\) stands for convergence in distribution, if and only if \[\label{eq:FW-cond} \int_{0}^1 \frac{1-g(s)}{f(s) - s} \mathrm{d}s < \infty.\tag{3}\] In this case the law of \(X_\infty\) is the unique stationary distribution of the Markov chain, see Corollary 1. Condition 3 holds in the subcritical case (\(\mathbf{E}A < 1\)) if and only if \(\mathbf{E}\log B < \infty\), this was proved by Quine [2] in the multitype setup. However, 3 can also hold in the critical case (\(\mathbf{E}A = 1\)), when necessarily \(\mathbf{E}A^2 = \infty\), see [1].

In the subcritical case, the tail behavior of the stationary law and the properties of the stationary process are well-studied. If either the offspring or the immigration distribution is regularly varying (with certain index range), then the tail behavior of the stationary law, and the properties of the stationary process were investigated by Basrak et al. [3]. More recently, Foss and Miyazawa [4] considered not only regularly varying tails, but more general tail behavior for the stationary law. In the random environment setup, Basrak and Kevei [5], and Kevei [6] obtained conditions ensuring the regular variation of the stationary law, and also investigated the properties of the stationary process.

Much less is known in the critical case. Recently, Guo and Hong [7] proved that if both the offspring and the immigration law is regularly varying then (under additional conditions on the slowly varying functions) the stationary law is also regularly varying. Zhao [8] obtained tail asymptotics of the stationary law in the borderline case, when the offspring distribution has tail \(\mathbf{P}( A > x ) \sim [x (\log x)^{1 + c}]^{-1}\), \(x \to \infty\), with \(c > 0\), and the immigration has tail \(\mathbf{P}( B > x ) \sim x^{-1}\), \(x \to \infty\). The properties of the stationary process are not studied. Here, and later on \(\sim\) stands for asymptotic equality, i.e. \(a(x) \sim b(x)\) as \(x \to L \in \{0,1,\infty\}\), if \(\lim_{x \to L} \tfrac{a(x)}{b(x)} = 1\). The same convention applies to sequences.

In the present paper we assume that the process is critical and, for some \(\alpha \in (0,1)\) and slowly varying function \(\ell_A\), \[\label{eq:f-ass} f(s) = s + (1-s)^{1+\alpha} \ell_A (1/(1-s)).\tag{4}\] On the immigration we assume that one of the following conditions holds:

• \(\mathbf{E}B < \infty\), or

• for some \(\beta \in (\alpha, 1)\) and slowly varying function \(\ell_B\), \[\label{eq:g-ass} g(s) := \mathbf{E}s^B = 1 - (1-s)^\beta \ell_B ( 1 / (1-s)).\tag{5}\]

It is easy to check that under these assumptions 3 holds, thus the stationary distribution exists.

Although there are not many results for processes with immigration, the critical branching process without immigration, where the offspring distribution has the form 4 , has attracted a lot of attention. In 1968, Slack [9] determined the extinction rate of the process, and obtained a Yaglom-type limit theorem in this setup. Borovkov and Vatutin [10] showed that the tail of the maximum of the process is \(c x^{-1}\), where the tail-index \(\alpha\) in 4 only appears in the multiplicative constant \(c\). In the same setup, Fleischmann et al. [11], among other results, obtained tail asymptotics for the total population.

Extending the results of Guo and Hong [7], in Theorem 1 we show that the tail of the stationary distribution is regularly varying. We further investigate the properties of the stationary process. The theory of heavy-tailed time series has attracted considerable mathematical attention recently, see e.g. the monographs by Mikosch and Wintenberger [12], and by Kulik and Soulier [13]. In Theorem 2 we determine the tail process of the stationary chain. The tail process, introduced by Basrak and Segers [14] reveals the behavior of the chain around its extremes. It turns out that the forward tail process is \(U_0 (1,1,\ldots)\), with \(U_0\) having Pareto distribution. This rather exotic behavior implies that the usual anti-clustering condition does not hold. The general theory of heavy-tailed time series works under mild conditions and implies point process convergence and convergence of partial sums and maxima. These mild conditions include the anti-clustering condition together with some mixing property. As a consequence, we cannot use the general theory. In Theorem 3 we prove a stable central limit theorem for the partial sums. Comparing the tail of the stationary law and the norming sequence, we obtain that the extremal index of the process is 0. Stationary time series with extremal index 0 are considered pathological. Apart from the Lindley process in queueing theory, we provide a further natural example of such a process.

Section 2 contains the results on the stationary distribution. In Section 3 we investigate the stationary process. All the proofs are gathered together in Section 4.

2 Stationary distribution↩︎

If the offspring generating function has the form 4 with some \(\alpha \in (0,1)\) and slowly varying function \(\ell_A\), and either (B1) or (B2) holds for the immigration generating function, then the Foster–Williamson condition 3 holds, thus \(X_n \stackrel{\mathcal{D}}{\to} X_\infty\), in particular a stationary distribution exists. From basic Markov chain theory we deduce that the stationary distribution is unique. For definitions on Markov chains we refer to Douc et al. [15]. Let \[\label{eq:def-k0} k_0 = \min \{ k: \, \mathbf{P}( B = k) > 0 ) \}.\tag{6}\] Then \(k_0\) is an accessible atom for the process \((X_n)\). Indeed, for any \(k> 0\) \[\mathbf{P}( X_{n+1} = k_0 | X_n = k ) = (\mathbf{P}( A=0))^k \mathbf{P}( B = k_0 ) > 0.\] Therefore, \((X_n)\) is irreducible. Furthermore, as stationary distribution exists, \((X_n)\) is recurrent, with a unique stationary distribution, see e.g. Theorems 7.1.4 and 7.2.1 in [15]. Let \(f_n\) denote the \(n\)-fold composition of the generating function \(f\), that is \(f_0(s) = s\), and \(f_{n+1}(s) = f_n(f(s))\), \(n \geq 1\). Then, by conditioning \[\mathbf{E}s^{X_{n+1}} = g(s) \mathbf{E}(f(s)^{X_n}) = g(s) g(f(s)) \mathbf{E}(f_2(s)^{X_{n-1}}) = \prod_{i=0}^n g(f_i(s)).\] We summarize this as follows.

Analyzing the asymptotics of the generating function in 7 , we determine the tail behavior of \(X_\infty\). For easier reference, we state a well-known result on the relation of the tail asymptotics at infinity and the generating function asymptotics at 1. We could not locate the precise statement in the literature, but it follows easily from a Tauberian theorem, see Corollary 8.1.7 in Bingham et al. [16], or Section 1.1 in [7].

We can state our main result on the tail of the stationary distribution.

In case (ii), under further assumption on the slowly varying functions, Guo and Hong [7] obtained the regular variation of the tail of the stationary distribution, without specifying the slowly varying function. In our result we do not assume further assumptions on the slowly varying functions, and we obtain explicit expression for the tail. Case (i) is completely new.

We also note that the result can be extended further to the boundary cases, when \(\alpha = 0\) or \(1\), or when \(\beta = \alpha\) or \(1\). In these boundary cases however, further assumptions are needed on the slowly varying functions in order to deduce the tail asymptotic. In particular, the corresponding slowly varying functions have to belong to the de Haan class, see [16] and the remark after it. See also our remarks after Lemmas 2 and 3. In order to keep the presentation simpler, we decided to exclude these cases.

3 The stationary Markov chain↩︎

Let \((X_n)_{n \in \mathbb{Z}}\) be the stationary process version of the Galton–Watson process with immigration, defined as \[\label{eq:X-def-stat} X_{n+1} = \sum_{i=1}^{X_n} A_{n+1,i} + B_{n+1} = \theta_{n+1} \circ X_n + B_{n+1}.\tag{9}\] In what follows we only need that the stationary distribution is regularly varying. Assume that for some \(\gamma \in (0,1)\) the tail of the stationary distribution \(X_\infty\) satisfies \[\label{eq:stat-ass} \mathbf{P}( X_\infty > x ) \sim \frac{\ell(x)}{x^{\gamma}}, \quad x \to \infty,\tag{10}\] where \(\ell\) is a slowly varying function. We showed that \(\gamma = 1-\alpha\) if 4 holds and \(\mathbf{E}B < \infty\), and \(\gamma = \beta- \alpha\) if 4 and 5 hold with \(1 \geq \beta > \alpha > 0\).

3.1 Tail process↩︎

The tail process of a stationary process \((X_n)_{n \in \mathbb{Z}}\), if it exists, is a process \((U_n)_{n \in \mathbb{Z}}\) such that for any \(k, \ell \geq 0\), as \(x \to \infty\) \[\mathcal{L} \left( x^{-1} (X_{-\ell}, X_{1-\ell}, \ldots, X_k) | X_0 > x \right) \stackrel{\mathcal{D}}{\longrightarrow} (U_{-\ell}, U_{-\ell + 1}, \ldots, U_k),\] and \(\mathbf{P}(U_0 > y) = y^{-\gamma}\), \(y \geq 1\), for some \(\gamma > 0\).

The tail process was introduced by Basrak and Segers [14], and became an essential tool in the analysis of heavy-tailed time series. It describes the behavior of the process \((X_n)\) around its extreme values. For definition and properties we refer to the original paper [14], and to the monographs [12] and [13]. The existence of the tail process is equivalent to the regular variation of the stationary process \((X_n)\). Furthermore, it is enough to prove the existence of one-sided tail process, that is for \(k=0, \ell \geq 0\), or for \(k \geq 0, \ell = 0\), see [12].

We note here that the Lindley process has the same tail process, see [12].

For a stationary regularly varying time series \((Y_n)\) choose \(a_n\) such that \(\mathbf{P}( Y_0 > a_n ) \sim n^{-1}\). The anti-clustering condition \(\mathcal{AC}(r_n, a_n)\) ([14], [13], [12]) holds, if for each \(u > 0\) \[\label{eq:antic} \lim_{m \to \infty} \limsup_{n \to \infty} \mathbf{P} \left( \max_{m \leq |k| \leq r_n} |Y_k| > a_n u \, \big| \, |Y_0| > a_n u\right) = 0,\tag{11}\] for some \(r_n\), with \(r_n \to \infty\) and \(r_n / n \to 0\) as \(n \to \infty\). Theorem 2 combined with Proposition 4.2 in [14] ([13]) implies that 11 does not hold for \((X_n)\). To apply the powerful machinery of heavy-tailed time series one needs the anti-clustering condition together with some weak form of mixing. Therefore, we cannot apply the general theory.

3.2 Critical Galton–Watson process without immigration↩︎

Here we recall some known results on critical Galton–Watson processes. Assume that for the offspring generating function 4 holds for some \(\alpha \in [0,1]\). For \(\alpha = 0\) we further assume that \(\lim_{x \to \infty} \ell_A(x) = 0\), which implies that \(\mathbf{E}A = 1\). Consider a Galton–Watson process with offspring generating function \(f\) as follows. Let \(Z_0 = 1\), and for \(n \geq 0\) \[\label{eq:Z-def} Z_{n+1} = \sum_{i=1}^{Z_n} A_{n+1, i},\tag{12}\] where \(\{ A_{n,i}: n\geq 1, i \geq 1 \}\) are iid random variables with generating function \(f\). The process is critical, therefore the total progeny is a.s. finite, \[\label{eq:T-def} T = Z_0 + Z_1 + \ldots .\tag{13}\] Let \(h(s) = \mathbf{E}s^T\) denote the generating function of \(T\). Then the functional equation \(h(s) = s f(h(s))\) easily implies tail asymptotics for \(T\). The following statement is Lemma 6 (or Theorem 2) in [11], see also [7]. In the current form, we allow also \(\alpha = 0\), which was excluded both in [11] and [7], and keep a precise track of the appearing slowly varying function. For completeness, we provide a short proof. First recall the notion of de Bruijn conjugate, see e.g. [16]. If \(\ell\) is a slowly varying function, there exists a slowly varying function \(\ell^{\#}\), unique up to asymptotic equivalence, such that \[\label{eq:deBruijn} \ell(x) \ell^{\#}(x \ell(x)) \to 1, \quad \ell^{\#}(x) \ell(x \ell^{\#}(x)) \to 1, \quad x \to \infty.\tag{14}\] Furthermore, \(\ell^{\# \#} \sim \ell\). In what follows, we often use these properties.

We note that for \(\alpha = 0\) the generating function asymptotics 15 does not imply 16 , only the regular variation of the truncated mean. For \(\alpha = 0\) the regular variation of \(\mathbf{P}( T > x)\) is equivalent to that \(1/\ell_{A,1}^{\#}\) belongs to the de Haan class \(\Pi\), see [16] and the remark after it.

3.3 Partial sum↩︎

We are interested in the asymptotic behavior of the partial sums \(S_n = \sum_{i=1}^n X_i\).

Note that for \(\alpha < 1\) the tail of the stationary distribution is regularly varying with index \(-(1-\alpha)\) in case of (B1), and \(-(\beta- \alpha)\) in case of (B2), which suggests larger scaling than in the theorem above. In particular, this also implies that the stable central limit theorem Theorem 9.2.1 in [12] (see also [13]) does not hold, hence either the mixing or the anti-clustering condition (different from 11 ) is not satisfied.

3.4 Maxima↩︎

For a stationary time series extremal dependence can be measured by the extremal index, introduced by Leadbetter [21], defined as follows, see Definition 6.1.7 in [12], or [13]. If for each \(\tau > 0\) there exists a sequence \(u_n(\tau)\) for which \[\label{eq:def-un} \lim_{n \to \infty} n \mathbf{P}( Y_0 > u_n(\tau) ) = \tau,\tag{17}\] such that \[\label{eq:def-theta} \lim_{n \to \infty} \mathbf{P}( M_n \leq u_n(\tau) ) = e^{-\theta_Y \tau},\tag{18}\] for some \(\theta_Y \in [0,1]\), where \(M_n = \max \{ Y_1, \ldots, Y_n \}\) stands for the partial maxima, then \(\theta_Y\) is the extremal index of \((Y_n)\). For an iid sequence \(\theta_Y =1\). For further properties we refer to [21] and [12].

Turning back to our stationary Markov chain \((X_n)\), let us assume that 10 holds. Then the sequence \(u_n(\tau)\) in 17 can be expressed as \[u_n(\tau) \sim (n/\tau)^{1/\gamma} \ell_1^{\#}(n)^{1/\gamma},\] where \(\ell_1(x) = 1/\ell(x^{1/\gamma})\). Simply, \(M_n \leq S_n\). By Theorems 1 and 3, in case (i) \(\gamma = 1-\alpha\) and \(\eta = 1/(1+\alpha)\), while in case (ii) \(\gamma = \beta - \alpha\) and \(\eta = \beta/(1+\alpha)\). In both cases \(u_n(\tau) / n^{1/\eta} \widetilde{\ell}(n) \to \infty\), implying that 18 holds with \(\theta_X =0\).

The extremal index \(\theta_X = 0\) is somewhat pathological, see Remark 6.1.10 in [12], and the discussion in [21]. There are only a few examples of such process, see e.g. [22]. A notable one is the Lindley process in queueing theory, see [12]. Thus we see that the extremal behavior of \((X_n)\) and the Lindley process is similar, as both have the same tail process and both have extremal index 0. For an intuitive meaning of \(\theta_X =0\), see the remark after Proposition 6.3.14 in [12].

Finally, we note that we expect that there is a proper normalization for the maxima \(M_n\). We showed in the beginning of Section 2 that the Markov chain has an accessible atom, therefore the idea of the proof of [12] could work. However, determining the tail behavior of the cycle maxima remains for further studies.

4 Proofs↩︎

4.1 Proof of Theorem 1↩︎

To ease notation, put \(\ell_1(x) = \ell_A(x^{1/\alpha}) ^{-1}\). By Lemma 2 in [9], \[(1-f_n(0))^\alpha \ell_A \left(\frac{1}{1-f_n(0)}\right) \sim \frac{1}{\alpha n}.\] Rewriting, we have \[(1- f_n(0))^{-\alpha} \ell_1((1-f_n(0))^{-\alpha}) \sim \alpha n,\] therefore, \[\label{eq:fn-asy} 1-f_n(0) \sim (\alpha n \ell_1^\#(\alpha n))^{-\tfrac{1}{\alpha}} \sim (\alpha n \ell_1^\#(n))^{-\tfrac{1}{\alpha}}.\tag{19}\] Since \(f_n(0) \uparrow 1\), for all \(s \in (f_1(0),1)\) there exists \(m(s)\) such that \[f_{m(s)}(0) < s < f_{m(s)+1}(0).\] Hence, \[1-f_{m(s)+1}(0) < 1-s < 1-f_{m(s)}(0),\] and by 19 , as \(s \uparrow 1\) \[\label{eq:m-asy0} 1-s \sim \left(\alpha m(s) \ell_1^\#(m(s)) \right)^{-1/\alpha}.\tag{20}\] It follows that \[\label{eq:m-asy} m(s) \sim \alpha^{-1} (1-s)^{-\alpha} \, \ell_1\left((1-s)^{-\alpha}\right), \quad s \uparrow 1.\tag{21}\] Furthermore, by 19 as \(n + m(s) \to \infty\) \[\label{eq:f95n} \begin{align} 1 - f_n(s) & \sim 1 - f_n(f_{m(s)}(0)) = 1 - f_{n+m(s)}(0) \\ & \sim (\alpha(n+m(s)))^{-1/\alpha} \, \ell_1^{\#}(n+m(s))^{-1/\alpha}. \end{align}\tag{22}\]

For any \(\varepsilon > 0\), there exists \(\delta(\varepsilon) > 0\) such that for \(x \in (0,\delta(\varepsilon))\) we have \(- ( 1 + \varepsilon) x \leq \log ( 1 -x) \leq - x\). Since \(f_n(s) \geq f_1(s) \geq s\), there exists \(s(\varepsilon) \in (0,1)\) such that \(g(s(\varepsilon)) \geq 1 - \delta(\varepsilon)\). Then for all \(n = 0,1,\ldots\), and \(s \in (s(\varepsilon),1)\) \[\label{eq:gf-aux1} - ( 1 + \varepsilon) ( 1 - g(f_n(s))) \leq \log g(f_n(s)) \leq - ( 1 - g(f_n(s))).\tag{23}\]

Case (B1): First, we suppose that the immigration has finite mean, \(\mathbf{E}B = g'(1) < \infty\). There exists \(y_\varepsilon \in (0,1)\) such that \((1-y) (1-\varepsilon) g'(1) \leq 1- g(y) \leq (1-y) g'(1)\) for \(y \in (y_\varepsilon, 1)\), we obtain from 23 and Corollary 1 that for \(s \geq \max\{ y_\varepsilon, s_\varepsilon \}\) \[\label{eq:varphi-asy-aux1} -(1 + \varepsilon) g'(1) \sum_{n=0}^\infty ( 1 - f_n(s)) \leq \log \varphi(s) \leq -(1 - \varepsilon) g'(1) \sum_{n=0}^\infty ( 1 - f_n(s)).\tag{24}\] As \(\varepsilon > 0\) is arbitrarily small, we obtain that \[\label{eq:varphi-asy} -\log \varphi(s) \sim g'(1) \sum_{n=0}^\infty ( 1 - f_n(s)) \quad \text{as } s \uparrow 1.\tag{25}\]

Fix \(N > 0\). Then, by 20 , if \(1-s\) is small enough \[\label{eq:phi-sum-aux1} \sum_{n=0}^{\lfloor m(s)/N \rfloor} (1 - f_n(s)) \leq \frac{m(s)}{N} (1- s) \leq \frac{2 \alpha^{-1/\alpha} }{N} m(s)^{1-1/\alpha} \ell_1^{\#}(m(s))^{-1/\alpha},\tag{26}\] and as \(s \uparrow 1\) \[\label{eq:phi-sum-aux2} \begin{align} & \sum_{n=\lfloor N m(s) \rfloor}^\infty (n+m(s))^{-1/\alpha} \ell_1^\#(n+m(s))^{-1/\alpha} \\ & \leq \sum_{n=\lfloor N m(s) \rfloor}^\infty n^{-1/\alpha} \, \ell_1^\#(n)^{-1/\alpha} \\ & \sim (N m(s))^{1-1/\alpha}\,\frac{1}{\tfrac{1}{\alpha} - 1} \, \ell_1^\#(N m(s))^{-1/\alpha} \\ & \sim \frac{\alpha}{1 - \alpha} N^{1 - 1/\alpha} \, m(s)^{1-1/\alpha} \, \ell_1^{\#}(m(s))^{-1/\alpha}. \end{align}\tag{27}\]

For the main contribution, by the uniform convergence theorem for slowly varying functions \[\label{eq:phi-sum-aux3} \begin{align} & \sum_{n=\lfloor m(s)/N \rfloor}^{\lfloor N m(s) \rfloor} (n+m(s))^{-1/\alpha} \ell_1^\#(n+m(s))^{-1/\alpha} \\ &= m(s)^{1-\frac{1}{\alpha}} \sum_{n=\lfloor \frac{m(s)}{N} \rfloor}^{\lfloor N m(s) \rfloor} \frac{1}{m(s)} \Big(\tfrac{n}{m(s)} + 1\Big)^{-1/\alpha} \ell_1^\#\!\big(m(s)(\tfrac{n}{m(s)} + 1)\big)^{-1/\alpha} \\ &\sim m(s)^{1-1/\alpha} \ell_1^\#(m(s))^{-1/\alpha} \int_{N^{-1}}^N (1 + y)^{-1/\alpha} \, \mathrm{d}y. \end{align}\tag{28}\] Hence, letting \(N \to \infty\), by 25 , 26 , 27 , and 28 we obtain \[\label{eq:varphi-asy2} \begin{align} - \log \varphi(s) & \sim g'(1) \alpha^{-1/\alpha} \frac{\alpha}{1-\alpha} m(s)^{1- 1/\alpha} \ell_1^\#(m(s))^{-1/\alpha}. \end{align}\tag{29}\] Finally, using 21 and that \(\ell_1^{\#}(x \ell_1(x)) \sim \ell_1(x)^{-1}\) we have \[m(s)^{1-1/\alpha} \ell_1^{\#}(m(s))^{-1/\alpha} \sim \alpha^{1/\alpha -1} (1-s)^{1-\alpha} \ell_1((1-s)^{-\alpha}).\] Substituting back into 29 \[- \log \varphi(s) \sim \frac{g'(1)}{1-\alpha} (1-s)^{1-\alpha} \frac{1}{\ell_A(1/(1-s))}, \quad s \uparrow 1.\] Since \(-\log \varphi(s) \sim 1- \varphi(s)\) as \(s \uparrow 1\), the statement follows from Lemma 1.

Case (B2): Now consider the case, when the immigration satisfies 5 with \(\beta \in (\alpha,1)\), and a slowly varying \(\ell_B\). Similarly, as before \[\begin{align} & - \log \varphi(s) \sim \sum_{n=0}^\infty (1-g(f_n(s))) \\ & \sim \sum_{n=0}^\infty (1-f_n(s))^\beta \, \ell_B\!\big((1-f_n(s))^{-1}\big) \\ & \sim \sum_{n=0}^\infty (1-f_{n+m(s)}(0))^\beta \, \ell_B\!\big((1-f_{n+m(s)}(0))^{-1}\big) \\ & \sim \sum_{n=0}^\infty \big( \alpha (n+m(s)) \, \ell_1^{\#}(n+m(s))\big)^{-\frac{\beta}{\alpha}} \ell_B \left( ((n+m(s)) {\ell_1^\#(n+m(s))})^{\frac{1}{\alpha}}\right) \\ &= \alpha^{-\beta/\alpha} \sum_{n=0}^\infty (n+m(s))^{-\beta/\alpha} \, \ell_2(n+m(s)), \end{align}\] where the asymptotic equality holds as \(s \uparrow 1\), and \[\ell_2(x) = (\ell_1^\#(x))^{-\beta/\alpha} \ell_B\left( (x \ell_1^{\#}(x))^{1/\alpha} \right).\] This is analogous to the previous case, with \(\alpha\) replaced by \(\alpha/\beta\) and \((\ell_1^{\#})^{-1/\alpha}\) by \(\ell_2\). Therefore, the same computation implies \[\label{eq:varphi2-asy} -\log \varphi(s) \sim \alpha^{-\beta/\alpha} \frac{\alpha}{\beta - \alpha} \, m(s)^{1-\beta/\alpha} \, \ell_2(m(s)), \quad s \uparrow 1.\tag{30}\] Noting that \(\ell_2(x \ell_1(x)) \sim \ell_1(x)^{\beta/\alpha} \ell_B(x^{1/\alpha})\), using 21 we have \[m(s)^{1-\beta/\alpha} \, \ell_2(m(s)) \sim \alpha^{\beta/\alpha -1} (1-s)^{\beta -\alpha} \ell_1((1-s)^{-\alpha}) \ell_B ((1-s)^{-1}).\] Substituting into 30 \[-\log \varphi(s) \sim \frac{1}{\beta - \alpha} (1-s)^{\beta -\alpha} \frac{\ell_B ((1-s)^{-1})}{\ell_A((1-s)^{-1}) }, \quad s \uparrow 1.\] The result follows again from Lemma 1.

4.2 Proofs of Theorem 2↩︎

The regular variation property implies that \(\mathcal{L}(X_0 / x | X_0 > x) \stackrel{\mathcal{D}}{\rightarrow} U_0\). As \(x_0 \to \infty\) by the law of large numbers a.s. \[\frac{1}{x_0} \left( \sum_{i=1}^{x_0} A_i + B_i \right) \longrightarrow 1.\] Therefore, as \(x \to \infty\) \[\mathcal{L}\left( \frac{X_1}{X_0} \Big| X_0 > x \right) \longrightarrow \delta_1,\] the latter being the degenerate distribution at 1. The statement follows by an induction argument.

4.3 Proof of Theorem 3↩︎

Before turning to the proof of Theorem 3 we need a result on the tail behavior of random sums, when both the number of summands and the summands are heavy-tailed with infinite mean. Part (i) is Proposition B.2.5 and part (ii) is Lemma B.2.7 in [12].

We note that in (i) the extra condition \(\mathbf{P}( \tau > x) / \mathbf{P}( Y > x) \to 0\) for \(\nu = 1\) is very weak, since \(x \mathbf{P}( \tau > x) \to 0\) always holds by \(\mathbf{E}\tau < \infty\). In general, the extra condition is needed, even if \(\mathbf{E}Y = \infty\) is assumed. It is clear from the proof that similar statement holds in (ii) allowing \(\mu = 1\) and \(\nu = 1\) if the appearing slowly varying function belongs to the de Haan class, see the remark after Lemma 2. For the sake of readability, we decided to exclude this case.

References↩︎