April 02, 2026
Multiplicative cascades model the successive fragmentation of a conserved quantity across scales and arise in fully developed turbulence [1], [2], rainfall, finance, and other settings exhibiting intermittent, scale-invariant fluctuations. The mathematical foundations of multiplicative cascades were established by Kahane and Peyrière [3]; see Barral and Mandelbrot [4] and Bacry and Muzy [5] for the modern discrete and continuous theories. The statistical properties of a cascade are encoded in the scaling exponents \(\zeta_p\) of the structure functions \(S_p(\ell) = \langle |\Phi(\ell)|^p \rangle \sim \ell^{\zeta_p}\). A central question is: which probability distributions on the cascade multiplier \(W\) are compatible with observed scaling laws?
Kolmogorov [1] proposed log-normal multipliers, leading to quadratic scaling exponents. Z.-S. She and Lévêque [6] introduced a hierarchical symmetry for the scaling exponents and derived a different, log-Poisson exponent formula that has since shown excellent agreement with experimental data. Dubrulle [7] independently identified the log-Poisson form. Z.-S. She and Waymire [8] used the Lévy–Khintchine representation to argue that this symmetry selects log-Poisson within the log-infinitely-divisible family; Dubrulle and Graner [9] reached a similar conclusion via symmetry groups. These works provided compelling physical arguments but did not supply rigorous proofs. Z.-S. She and Zhang [10] subsequently proposed that the hierarchical symmetry is universal—applicable not only to turbulence but to general multi-scale fluctuation systems including MHD turbulence, natural image statistics, and biological signals—and should serve as a standard analytical framework. The present paper supplies the rigorous mathematical foundation for this program.
We formalize the hierarchical symmetry as a single axiom (A1) and prove the following.
Characterization (Theorem 1). A1 uniquely determines the cascade multiplier \(W\) to be log-Poisson, with parameters \((\,a = \gamma\ln r,\; b = (\ln\beta)/k,\; \lambda = -C\ln r\,)\) expressed in terms of the observable scaling exponents. No other distribution—infinitely divisible or not—is compatible with A1.
Classification (Theorem 3). Within the full log-infinitely-divisible family, A1 selects exactly the log-Poisson class, and the proof stratifies the exclusion: Gaussian components, positive jumps, and stable generators of index \(\alpha \in [1,2)\) are eliminated by divergence of the incremental exponents; all remaining generators are eliminated by a second-moment rigidity.
Stability (Theorem 5). If A1 holds only approximately, with residuals bounded by \(\varepsilon\), then the (tilted, compactified) Lévy measure of the generator is within \(K\sqrt{\varepsilon}\) of a Dirac mass in the Wasserstein-1 metric, with the sharp constant \(K = ((1+\beta)|\ln r|/|A|)^{1/2}\) when the limiting increment \(\delta_\infty\) is known, and the sharp constant \((2|\ln r|/|A|)^{1/2}\) when \(\delta_\infty\) is fitted along with \(\beta\).
Propagation at the sharp rate (Theorems 6 and 7). Closeness of the Lévy measure transfers to closeness of the multiplier distribution at rate \(O(\sqrt\varepsilon)\) unconditionally—no finite-activity or minimum-jump hypothesis—with an explicit constant; and an explicit family shows the rate \(\Theta(\sqrt\varepsilon)\) is exact.
Beyond independence (Section 6). For stationary ergodic multipliers, A1 is equivalent to the limiting cumulant generating function—hence the large-deviation rate and the multifractal spectrum—being exactly log-Poisson (Theorem 8); but it provably does not determine the multiplier law: an explicit stationary ergodic countable-state Markov multiplier satisfies A1 exactly with a non-log-Poisson marginal (Theorem 9). Exchangeable multipliers with exact scaling collapse to the i.i.d. log-Poisson cascade (Corollary 4), and finite-state Markov multipliers cannot satisfy A1 at all with nontrivial intermittency (Theorem 10).
Continuous cascades (Section 7). In the Bacry–Muzy category of exactly scale-invariant log-infinitely-divisible multifractal random measures, structure-function exponents exist only on a finite moment window, and no finite window identifies the cascade class (Theorem 11); at the level of the scale-invariance generator—magnitude statistics, observable at all orders—A1 selects exactly the Barral–Mandelbrot compound Poisson cascade (Theorem 12), with stability constants that are native and scale-ratio-free (Corollary 5).
The converse—that log-Poisson multipliers imply A1—is established in Proposition 2, yielding a biconditional equivalence (Corollary 2).
Relation to prior work. The exponent formula ?? (Lemma 1) and the log-Poisson identification are due to Z.-S. She and Lévêque [6] and Dubrulle [7]. Z.-S. She and Waymire [8] gave the first argument connecting A1 to the Lévy–Khintchine classification. The compound Poisson cascades appear in Barral–Mandelbrot [4]; the log-ID multifractal random measures in Bacry–Muzy [5], [11]; magnitude-cumulant analysis in Delour–Muzy–Arneodo [12]. The following results are new: the boundedness and moment-determinacy lemma (Lemma 2); the converse and biconditional (Proposition 2, Corollary 2); the classification with stratified exclusion (Theorem 3); the determinacy dichotomy (Proposition 4); the stability theorem with sharp constants under both readings (Theorem 5); the unconditional sharp-rate propagation theory (Theorems 6, 7); the beyond-i.i.d.classification and its impossibility boundary (Theorems 8, 9, 10, Corollary 4); and the continuous-cascade results (Theorems 11, 12).
Method. The change of variables \(u = e^{kx}\) maps the Lévy measure from \((-\infty,0]\) to the compact interval \([0,1]\), where a second-moment identity against the candidate atom decides the classification and its stability. A1 forces the multiplier to be essentially bounded with \(\mathop{\mathrm{ess\,sup}}W = r^\gamma\), which yields moment determinacy directly and eliminates all tail estimates from the propagation argument; the propagation itself is a multiplicative coupling in which jumps near the identity are costed by their multiplicative deviation rather than counted. Beyond independence the arguments run at the level of limiting cumulant generating functions; in the continuous category they transfer through the dictionary \(\ln r \mapsto -1\).
Throughout this paper, \(\log\) denotes the natural logarithm.
Let \(r \in (0,1)\) be a scale ratio. A multiplicative cascade generates a random positive measure \(\mu\) on nested sets \(B_0 \supset B_1 \supset \cdots\) via \[\mu(B_{n+1}) = W_{n+1} \cdot \mu(B_n),\] where \(\{W_n\}\) are i.i.d.positive random variables with \(\mathbb{E}[W] = 1\) (conservation of mean; a normalization convention—none of the proofs below depend on it, cf. Corollary 6). Sections 6 and 7 relax, respectively, the independence assumption and the discrete setting.
Let \(\Phi(\ell)\) be the cascade observable at scale \(\ell = r^n\). Define the structure functions \[S_p(\ell) = \langle |\Phi(\ell)|^p \rangle = \ell^{\zeta_p},\] where \(\zeta_p\) are the scaling exponents, with \(\zeta_0 = 0\); all moments of \(W\) are assumed finite, so that \(\zeta_p\) is finite for every \(p \ge 0\). The relation \(\mathbb{E}[W^p] = r^{\zeta_p}\) identifies the single-step moment structure of the multiplier with the observable’s scaling.
For a fixed integer \(k \geq 1\) (the hierarchy step), define the moment ratios \[H_p(\ell) = \frac{S_{p+k}(\ell)}{S_p(\ell)} = \ell^{\delta_p},\] where \(\delta_p = \zeta_{p+k} - \zeta_p\) are the incremental exponents at step \(k\).
Axiom 1 (A1: Hierarchical Symmetry). There exist \(\beta \in (0,1)\) and \(L \in \mathbb{R}\) such that for all \(p \in k\mathbb{N}_0\) (non-negative integer multiples of \(k\)), the incremental exponents satisfy \[\begin{align} \label{eq:A1} \delta_{p+k} = (1 - \beta)\,L + \beta\,\delta_p. \end{align}\qquad{(1)}\]
Since ?? is a contraction, it forces \(\delta_{mk} \to L\) as \(m \to \infty\); we write \(\delta_\infty := L\). (Stating the axiom with a free constant \(L\), rather than defining \(\delta_\infty\) as a limit inside the equation that uses it, matters only for the approximate version in Theorem 5, where the distinction between the true limit and a fitted constant is quantitatively significant.)
A1 determines the full parameter set from the observable exponents:
| Parameter | Determined by | Meaning |
|---|---|---|
| \(\beta\) | Contraction ratio of ?? | Coupling strength |
| \(\gamma\) | \(\delta_\infty / k\) | Linear drift |
| \(C\) | \((\delta_0 - \delta_\infty)/(1 - \beta)\) | Concentration amplitude |
Edge case (\(C = 0\)). If \(\delta_0 = \delta_\infty\), then \(C = 0\) and \(\zeta_p = \gamma p\) (monofractal scaling). The cascade multiplier \(W = r^\gamma\) is deterministic. A1 is trivially satisfied for any \(\beta \in (0,1)\). The classification and stability theorems assume \(C > 0\) (nontrivial intermittency).
Lemma 1 (Exponent Form). If the incremental exponents \(\{\delta_p\}\) satisfy the A1 recurrence ?? with \(\beta \in (0,1)\), then with \(\zeta_0 = 0\): \[\begin{align} \label{eq:zeta} \zeta_p = \gamma p + C\bigl(1 - \beta^{p/k}\bigr), \end{align}\qquad{(2)}\] where \(\gamma = \delta_\infty / k\) and \(C = (\delta_0 - \delta_\infty)/(1 - \beta)\).
We emphasize that this lemma is purely algebraic and involves no probabilistic content.
Proof. (1) The recurrence ?? is first-order linear with fixed point \(\delta_\infty\): \[\delta_{p+k} - \delta_\infty = \beta\,(\delta_p - \delta_\infty).\]
(2) At \(p = mk\) (integer multiples of \(k\)), iteration gives \[\delta_{mk} - \delta_\infty = (\delta_0 - \delta_\infty)\,\beta^m.\] Replacing \(m = p/k\): \[\delta_p = \delta_\infty + (\delta_0 - \delta_\infty)\,\beta^{p/k}.\] (This formula is derived at \(p \in k\mathbb{N}_0\). For general \(p \geq 0\), we define \(\zeta_p\) by ?? ; the function \(\beta^{p/k}\) is well-defined for all real \(p \geq 0\) since \(\beta > 0\). The moment-based arguments in Lemma 2 and Theorem 1 use only the lattice \(p \in k\mathbb{N}_0\), where the formula is proved.)
(3) With \(\zeta_0 = 0\), sum the step-\(k\) increments using the geometric series: \[\zeta_p = \frac{p}{k}\,\delta_\infty + \frac{\delta_0 - \delta_\infty}{1 - \beta}\, \bigl(1 - \beta^{p/k}\bigr).\] Identifying \(\gamma = \delta_\infty/k\) and \(C = (\delta_0 - \delta_\infty)/(1-\beta)\): \[\zeta_p = \gamma p + C\bigl(1 - \beta^{p/k}\bigr). \qedhere\] ◻
Lemma 2 (Boundedness and Moment Determinacy). Let \(\{W_n\}\) be an i.i.d.multiplicative cascade whose lattice scaling exponents satisfy \(\zeta_{km} = \gamma km + C(1-\beta^{m})\) for all \(m \in \mathbb{N}_0\), as produced by A1 via Lemma 1. Then:
(i) \(W\) is essentially bounded, with \(\mathop{\mathrm{ess\,sup}}W = r^{\gamma}\);
(ii) the law of \(W\) is uniquely determined by the lattice moments \(\{\mathbb{E}[W^{km}]\}_{m \ge 0}\).
Proof. Set \(V = W^k \ge 0\). By independence across cascade levels, \(\mathbb{E}[(W_1\cdots W_n)^p] = (\mathbb{E}[W^p])^n\); combined with \(S_p(\ell) = r^{n\zeta_p}\) this gives, for a single step, \(\mathbb{E}[V^m] = \mathbb{E}[W^{km}] = r^{\zeta_{km}}\) for every \(m \in \mathbb{N}_0\)—these are exactly the moments constrained by A1. Then \[\bigl(\mathbb{E}[V^m]\bigr)^{1/m} \;=\; r^{\zeta_{km}/m} \;=\; r^{\,\gamma k + C(1-\beta^m)/m} \;\longrightarrow\; r^{\gamma k} \qquad (m \to \infty).\] On a probability space the norms \(\|V\|_{L^m}\) are nondecreasing in \(m\) and converge to \(\|V\|_{L^\infty}\); hence \(V\) is essentially bounded with \(\|V\|_\infty = r^{\gamma k}\), and \(W = V^{1/k}\) is bounded with \(\|W\|_\infty = r^{\gamma}\), proving (i).
For (ii): a probability law supported in the compact interval \([0, r^{\gamma k}]\) is uniquely determined by its integer moments. (Hausdorff moment problem: polynomials are uniformly dense in \(C([0,r^{\gamma k}])\) by the Weierstrass approximation theorem, so two laws with equal moments integrate every continuous function equally and coincide by the Riesz representation theorem.) Hence the law of \(V\) is determined by \(\{\mathbb{E}[V^m]\}\); since \(x \mapsto x^{1/k}\) is a Borel bijection of \([0,\infty)\), the law of \(W\) is determined as well. ◻
Remark 1. A Carleman-condition argument [13] is available here only along the lattice (applied to \(V = W^k\)), since A1 constrains no other moments when \(k \ge 2\). The boundedness route above is shorter and stronger: statement (i) identifies the essential supremum of the multiplier with the most-singular scaling factor \(r^\gamma\), a fact used again in Theorem 6.
Theorem 1 (Characterization). Let \(\{W_n\}\) be an i.i.d.multiplicative cascade whose incremental scaling exponents satisfy A1. Then:
(i) Scaling exponents. \[\zeta_p = \gamma p + C\bigl(1 - \beta^{p/k}\bigr),\] where \(\gamma = \delta_\infty/k\) and \(C = (\delta_0 - \delta_\infty)/(1-\beta)\).
(ii) Uniqueness. The cascade multiplier \(W\) is uniquely determined to be log-Poisson: \[\log W = a + bN, \qquad N \sim \mathrm{Poisson}(\lambda),\] \[a = \gamma \ln r, \quad b = \frac{\ln\beta}{k}, \quad \lambda = -C\ln r.\] No other probability distribution on \(W\) is compatible with A1.
(iii) Multifractal spectrum. Let \(d\) denote the spatial dimension of the cascade support. Then \[f(h) = d - C + Cx(1 - \ln x), \qquad x = \frac{k(h - \gamma)}{C|\ln\beta|},\] defined for \(h \in [\gamma,\; \gamma + (C/k)|\ln\beta|]\).
Proof. (i) Immediate from Lemma 1.
(ii) Set \(a = \gamma\ln r\), \(b = (\ln\beta)/k\), \(\lambda = -C\ln r > 0\). For \(\log W = a + bN\) with \(N \sim \mathrm{Poisson}(\lambda)\): \[\mathbb{E}[W^p] = e^{ap} \cdot \exp\bigl[\lambda(e^{bp} - 1)\bigr] \qquad\text{for all real } p \ge 0,\] and since \(e^{bp} = \beta^{p/k}\), the choice \(\lambda = -C\ln r\) gives \(\lambda(\beta^{p/k}-1) = C\ln r\,(1-\beta^{p/k})\), so that \[\mathbb{E}[W^{km}] = e^{(\gamma\ln r) km}\, e^{C\ln r\,(1-\beta^{m})} = r^{\zeta_{km}} \qquad\text{for every } m \in \mathbb{N}_0 .\] Thus the log-Poisson law realizes exactly the lattice moments of the cascade multiplier. By Lemma 2(ii) the lattice moments uniquely determine the law; hence \(W\) is log-Poisson, and no other distribution is possible. (The same matching holds at every real \(p \ge 0\), so the extension of ?? off the lattice is consistent.)
(iii) The singularity spectrum \(f(h) = \inf_p\,[ph - \zeta_p + d]\). Setting the derivative to zero: \[h - \gamma + \frac{C}{k}(\ln\beta)\,\beta^{p/k} = 0 \quad\Longrightarrow\quad h - \gamma = \frac{C|\ln\beta|}{k}\,\beta^{p/k}.\] Define \(x = \beta^{p/k} = k(h-\gamma)/(C|\ln\beta|)\). Then \(p = -k\ln x / |\ln\beta|\) and \[p(h - \gamma) = \frac{-k\ln x}{|\ln\beta|} \cdot \frac{C|\ln\beta|}{k}\,x = -Cx\ln x.\] Therefore \[f(h) = d - C + Cx(1 - \ln x), \qquad x = \frac{k(h-\gamma)}{C|\ln\beta|}.\] Boundary checks: \(p = 0 \Rightarrow x = 1 \Rightarrow f = d\); \(p \to \infty \Rightarrow x \to 0 \Rightarrow f \to d - C\). Concavity: \(\zeta_p'' = -(C/k^2)(\ln\beta)^2\beta^{p/k} < 0\). ◻
Corollary 1 (Most-singular branch). Under A1, \(W \le r^\gamma\) almost surely, and the bound is attained with positive probability: \(\mathbb{P}(W = r^\gamma) = e^{-\lambda}\). The probability that a cascade trajectory takes the maximal factor for \(n\) consecutive levels is \[e^{-\lambda n} \;=\; r^{Cn} \;=\; \ell^{\,C} \qquad\text{at scale } \ell = r^n :\] the set of always-maximal cascade paths carries codimension exactly \(C\), in agreement with \(f(h_{\min}) = d - C\) in Theorem 1(iii).
Proof. By Theorem 1(ii), \(W = r^\gamma \beta^{N/k}\) with \(N \sim \mathrm{Poisson}(\lambda)\), so \(W \le r^\gamma\) with equality iff \(N = 0\), an event of probability \(e^{-\lambda}\). By independence across levels, \(n\) consecutive maximal factors have probability \(e^{-\lambda n} = e^{(C\ln r) n} = r^{Cn}\). ◻
Remark 2 (Scope of Theorem 1). No infinite-divisibility assumption enters Theorem 1: A1 characterizes the log-Poisson law within the class of all* nonnegative multipliers with finite lattice moments. Theorem 3 below is therefore not a larger uniqueness statement but an anatomical one: it locates the log-Poisson class inside the Lévy–Khintchine parameterization and identifies the mechanism by which each rival sub-family fails A1. Its proof technique—the compactifying substitution \(u = e^{kx}\)—is also the engine of the stability and propagation theory.*
Remark 3 (Conservation). The setup assumes \(\mathbb{E}[W] = 1\), which requires \(\zeta_1 = 0\). Substituting into ?? : \(\gamma + C(1 - \beta^{1/k}) = 0\), giving \(\gamma = -C(1-\beta^{1/k})\). This is a constraint relating \(\gamma\) to \(C\) and \(\beta\), reducing the free parameters from three to two.
Proposition 2 (Converse). If the cascade multiplier \(W\) is log-Poisson—that is, \(\log W = a + bN\) with \(N \sim \mathrm{Poisson}(\lambda)\), \(b < 0\), \(\lambda > 0\)—then the incremental scaling exponents satisfy A1 with \(\beta = e^{bk} \in (0,1)\).
Proof. The moment generating function gives \(\mathbb{E}[W^p] = \exp(ap + \lambda(e^{bp} - 1))\), so \(\zeta_p = \bigl(ap + \lambda(e^{bp} - 1)\bigr)/\ln r\). The step-\(k\) increments are \[\delta_p = \frac{ak + \lambda e^{bp}(e^{bk}-1)}{\ln r}.\] Setting \(\beta = e^{bk} \in (0,1)\) (since \(b < 0\), \(k \geq 1\)): \[\delta_p = \frac{ak}{\ln r} + \frac{\lambda(\beta - 1)}{\ln r}\,\beta^{p/k}.\] As \(p \to \infty\): \(\beta^{p/k} \to 0\), so \(\delta_\infty = ak/\ln r\). The deviation is \[\delta_p - \delta_\infty = \frac{\lambda(\beta - 1)}{\ln r}\,\beta^{p/k}.\] At \(p + k\): \[\delta_{p+k} - \delta_\infty = \frac{\lambda(\beta-1)}{\ln r}\,\beta^{(p+k)/k} = \beta\,(\delta_p - \delta_\infty).\] Therefore \(\delta_{p+k} = (1-\beta)\delta_\infty + \beta\delta_p\), which is exactly A1. ◻
Corollary 2 (Biconditional). Within i.i.d.multiplicative cascades, A1 is necessary and sufficient for log-Poisson: \[\text{A1 holds} \;\;\Longleftrightarrow\;\; W \text{ is log-Poisson (with } b < 0\text{).}\] The forward direction is Theorem 1(ii); the reverse is Proposition 2.
Theorem 3 (Log-ID Classification). Let \(\{W_n\}\) be an i.i.d.multiplicative cascade with nontrivial intermittency (\(C > 0\)), whose generator \(\log W\) is infinitely divisible with Lévy triplet \((a, \sigma^2, \nu)\). Then A1 holds with \(\beta \in (0,1)\) if and only if \(\sigma^2 = 0\) and \(\nu = \lambda\delta_b\) for some \(b < 0\), \(\lambda > 0\). That is:
A1 selects exactly the log-Poisson class from the full log-infinitely-divisible family.
No other log-ID cascade—log-normal, log-stable, or any intermediate—satisfies A1.
Proof. Reverse direction. If \(\nu = \lambda\delta_b\) with \(b < 0\) and \(\sigma^2 = 0\), then \(\log W = a + bN\) with \(N \sim \mathrm{Poisson}(\lambda)\), and A1 holds by Proposition 2.
Forward direction. Assume A1 holds. We show \(\sigma^2 = 0\) and \(\nu = \lambda\delta_b\).
Step 1 (unsplit form). The cumulant generating function of \(\log W\) is \[\psi(p) = ap + \frac{\sigma^2 p^2}{2} + \int\bigl(e^{px} - 1 - px\,\mathbb{1}_{|x|\leq 1}\bigr) \,\nu(dx),\] finite for all \(p \ge 0\) since all moments of \(W\) are finite. With \(\zeta_p = \psi(p)/\ln r\) and \(\delta_p = (\psi(p+k) - \psi(p))/\ln r\), define \(\phi(p) = \psi(p+k) - \psi(p)\). Then \[\begin{align} \phi(p) &= ak + \sigma^2 k\!\left(p + \tfrac{k}{2}\right) + \int g_p(x)\,\nu(dx),\\ g_p(x) &:= e^{px}\bigl(e^{kx}-1\bigr) - kx\,\mathbb{1}_{|x|\leq 1}, \end{align}\] where \(g_p\) is \(\nu\)-integrable for each \(p\), being the difference of the two compensated Lévy–Khintchine integrands. No splitting of the integral is performed at this stage.
Step 1\('\) (sign inventory). For every \(p \ge 0\):
on \((0,1]\): \(e^{px} \ge 1\) gives \(g_p(x) \ge (e^{kx}-1) - kx \ge 0\), and \(g_p(x) \uparrow \infty\) pointwise as \(p \to \infty\);
on \((1,\infty)\): \(g_p(x) = e^{px}(e^{kx}-1) \ge 0\), increasing to \(+\infty\) pointwise;
on \([-1,0)\): \(|e^{px}(e^{kx}-1)| \le 1 - e^{kx} \le k|x|\), so \(0 \le g_p(x) \le k|x|\), with \(g_p(x) \to k|x|\) pointwise as \(p \to \infty\);
on \((-\infty,-1)\): \(-1 \le g_p(x) \le 0\), with \(g_p(x) \to 0\) pointwise.
In particular \(\int g_p\,d\nu \ge -\nu((-\infty,-1))\), uniformly in \(p\).
Step 2 (\(\sigma^2 = 0\)). If \(\sigma^2 > 0\) then, by Step 1\('\), \[\phi(p) \;\ge\; ak + \sigma^2 k\Bigl(p + \tfrac{k}{2}\Bigr) - \nu\bigl((-\infty,-1)\bigr) \;\longrightarrow\; +\infty,\] so \(\delta_p = \phi(p)/\ln r \to -\infty\), contradicting the finite limit \(\delta_\infty\) forced by A1. Hence \(\sigma^2 = 0\). This eliminates all log-normal and mixed Gaussian-jump generators.
Step 3 (\(\mathop{\mathrm{supp}}(\nu) \subseteq (-\infty,0]\)). If \(\nu\) has mass on \((0,\infty)\) then, since \(g_p \ge 0\) there and \(g_p \uparrow \infty\) pointwise, monotone convergence gives \(\int_{(0,\infty)} g_p\,d\nu \to \infty\), while \(\int_{(-\infty,0)} g_p\,d\nu \ge -\nu((-\infty,-1))\); again \(\phi(p) \to \infty\) and \(\delta_p \to -\infty\), a contradiction. Therefore \(\mathop{\mathrm{supp}}(\nu) \subseteq (-\infty,0]\). This eliminates all generators with positive jumps.
Step 3½ (integrability near \(0\)). We claim A1 forces \(\int_{[-1,0)} |x|\,\nu(dx) < \infty\). With \(\sigma^2 = 0\) and \(\mathop{\mathrm{supp}}\nu \subseteq (-\infty,0]\), apply Fatou’s lemma on \([-1,0)\) (integrand \(g_p \ge 0\) by Step 1\('\), pointwise limit \(k|x|\)) and dominated convergence on \((-\infty,-1)\) (bounded by \(1\), finite mass): \[\liminf_{p\to\infty} \phi(p) \;\ge\; ak + k\int_{[-1,0)} |x|\,\nu(dx) - \nu\bigl((-\infty,-1)\bigr).\] If \(\int_{[-1,0)}|x|\,d\nu = \infty\) then \(\phi(p) \to \infty\) and \(\delta_p \to -\infty\), contradicting A1. Hence \(\int_{|x|\le1}|x|\,d\nu < \infty\), the compensator integral \(k\int x\,\mathbb{1}_{|x|\le1}\,d\nu\) is finite, and only now may the integral be split: \[\phi(p) = c_0 + \int_{(-\infty,0)} e^{px}\bigl(e^{kx}-1\bigr) \,\nu(dx), \qquad c_0 = ak - k\!\int x\,\mathbb{1}_{|x|\leq 1}\,\nu(dx).\] The remaining integrand is dominated by \(1 - e^{kx} \le \min(1, k|x|) \in L^1(\nu)\) and tends to \(0\) pointwise, so by dominated convergence \(\phi(p) \to c_0\) as \(p \to \infty\); thus \(\phi_\infty = c_0\) and \(\delta_\infty = c_0/\ln r\).
Step 4 (\(\nu\) is a single Dirac mass). A1 at \(p = mk\) gives, by Lemma 1(2), \(\delta_{mk} - \delta_\infty = (\delta_0-\delta_\infty)\beta^m\); multiplying by \(\ln r\), \[\label{eq:dagger} \int_{(-\infty,0)} e^{mkx}\bigl(e^{kx}-1\bigr)\,\nu(dx) = A\beta^m \qquad\text{for all } m \geq 0,\tag{1}\] where \(A = (\delta_0 - \delta_\infty)\ln r < 0\) (nontrivial intermittency and \(\ln r < 0\)). Substitute \(u = e^{kx}\), mapping \((-\infty,0) \to (0,1)\); let \(\tilde{\nu}\) be the pushforward of \(\nu\) and set \[\eta := (1-u)\,d\tilde{\nu} \;\ge\; 0, \qquad \mu_m := \int_{(0,1)} u^m \, d\eta .\] Then 1 reads \(\mu_m = |A|\,\beta^m\) for all \(m \ge 0\); the case \(m=0\) shows \(\eta\) is a finite positive measure of total mass \(|A|\). Only \(m \in \{0,1,2\}\) are needed: \[\int_{(0,1)} (u-\beta)^2 \, d\eta \;=\; \mu_2 - 2\beta\mu_1 + \beta^2\mu_0 \;=\; |A|\bigl(\beta^2 - 2\beta^2 + \beta^2\bigr) \;=\; 0 .\] Since \((u-\beta)^2 > 0\) on \((0,1)\setminus\{\beta\}\) and \(\eta \ge 0\), we conclude \(\eta\bigl((0,1)\setminus\{\beta\}\bigr) = 0\) and \(\eta(\{\beta\}) = |A|\). Because \(u - 1 \neq 0\) on \((0,1)\), the tilt is invertible: \[\tilde{\nu} = \frac{|A|}{1-\beta}\,\delta_\beta = \lambda\,\delta_\beta, \qquad \lambda = \frac{|A|}{1-\beta} > 0 .\] Therefore \(\nu = \lambda\delta_b\) with \(b = (\ln\beta)/k < 0\) and \(\lambda > 0\). The generator \(\log W\) is compound Poisson with deterministic jump size \(b\) and rate \(\lambda\): this is the log-Poisson distribution. ◻
Remark 4 (Alternative identification; minimality). The conclusion of Step 4 can also be reached from the full moment sequence: a finite signed measure on a compact interval is determined by its moments (Weierstrass approximation and the Riesz representation theorem), and \(A\delta_\beta\) realizes the moments 1 . The second-moment argument given above is preferred because it (a) uses only \(m \in \{0,1,2\}\) of 1 , so that A1 restricted to \(p \in \{0, k, 2k\}\), together with Steps 2–3½, already pins the distribution; and (b) is exactly the computation that the stability theorem quantifies (see the Remark closing Section 5).
Remark 5 (Which families die where). The exclusion mechanism stratifies. Gaussian components (Step 2), positive jumps (Step 3), and negative-support Lévy measures with \(\int_{|x|\le1}|x|\,d\nu = \infty\)—in particular totally skewed stable generators of index \(\alpha \in [1,2)\)—all fail A1 by divergence: \(\delta_\infty = -\infty\) (Step 3½). All remaining log-ID generators have bounded incremental exponents but fail the geometric rigidity* of Step 4: e.g.for a stable generator of index \(\alpha < 1\) the moments \(\mu_m\) decay like the power law \(m^{\alpha-1}\), which cannot equal \(|A|\beta^m\) for any \(\beta \in (0,1)\).*
Corollary 3 (Principal cascade classes). The log-ID cascade family is partitioned by A1:
| Class | Lévy data | A1 | Failure mode | Determinate | ||
|---|---|---|---|---|---|---|
| Log-Poisson |
|
Holds | — | Yes (bounded \(W\)) | ||
| Log-normal | \(\sigma^2>0\) | Fails |
|
No | ||
| Log-stable, \(\alpha\in[1,2)\) | \(\nu\) power-law | Fails |
|
— | ||
| Log-stable, \(\alpha<1\) | \(\nu\) power-law | Fails |
|
Yes (bounded \(W\)) | ||
| General log-ID | any other | Fails | Step 3 or Step 4 | — |
Determinacy in this family tracks boundedness of the multiplier, equivalently boundedness of \(\{\delta_p\}\) (Lemma 2(i)): every negative-support generator with finite \(\delta_\infty\) has compactly supported \(W\), hence is moment-determinate. The operative dichotomy is bounded versus unbounded, with A1 strictly on the bounded side.
Proposition 4 (Determinacy Dichotomy). The two principal cascade exponent laws are distinguished by moment determinacy:
(a) If A1 holds within a cascade (log-Poisson regime), then the scaling exponents uniquely determine the multiplier law (Theorem 1(ii)); indeed \(W\) is bounded and moment-determinate (Lemma 2).
(b) If the exponents are quadratic, \(\zeta_p = c_1 p + c_2 p^2\) with \(c_2 < 0\) ([1]/log-normal regime), then \(\mathbb{E}[W^p] = \exp(\mu p + \sigma^2 p^2/2)\) with \(\mu = c_1\ln r\) and \(\sigma^2 = 2c_2\ln r > 0\), and this moment sequence is indeterminate: uncountably many distinct laws realize it. Under quadratic scaling, the exponents cannot identify the multiplier law.
Proof. Part (a) is contained in Lemma 2 and Theorem 1(ii).
Part (b). We exhibit the family (Heyde [14]). Let \(f\) be the log-normal density with parameters \((\mu, \sigma^2)\) and, for \(|c| \le 1\), define \[f_c(x) \;=\; f(x)\,\Bigl[\,1 + c\, \sin\!\Bigl(\tfrac{2\pi(\ln x - \mu)}{\sigma^2}\Bigr)\Bigr], \qquad x > 0 .\] Substituting \(\ln x = \mu + \sigma z\) with \(Z\) standard normal, for every \(n \in \mathbb{N}_0\): \[\begin{align} \int_0^\infty x^n f(x) \sin\!\Bigl(\tfrac{2\pi(\ln x-\mu)}{\sigma^2}\Bigr)\,dx &= e^{n\mu}\,\mathbb{E}\Bigl[e^{n\sigma Z} \sin\!\Bigl(\tfrac{2\pi Z}{\sigma}\Bigr)\Bigr]\\ &= e^{n\mu}\, e^{(n^2\sigma^2 - 4\pi^2/\sigma^2)/2}\, \sin(2\pi n) = 0, \end{align}\] using \(\mathbb{E}[e^{(\alpha+i\beta')Z}] = e^{(\alpha+i\beta')^2/2}\), whose imaginary part is \(e^{(\alpha^2-\beta'^2)/2}\sin(\alpha\beta')\), with \(\alpha = n\sigma\), \(\beta' = 2\pi/\sigma\), \(\alpha\beta' = 2\pi n\). The case \(n = 0\) shows each \(f_c\) is a probability density (and \(f_c \ge 0\) since \(|c| \le 1\)); the cases \(n \ge 1\) show all \(f_c\) share the log-normal moments \(\exp(n\mu + n^2\sigma^2/2)\). Hence uncountably many distinct laws realize the moment sequence. ◻
Remark 6. Indeterminacy cannot be inferred from the convergence of the Carleman sum: Carleman’s condition is sufficient for determinacy but not necessary, so its failure proves nothing. The explicit family above is the classical argument.
Theorem 5 (Stability). Let \(\{W_n\}\) be an i.i.d.multiplicative cascade with log-infinitely-divisible generator, all moments finite, and nontrivial intermittency. Suppose there exist \(\beta \in (0,1)\), \(d^* \in \mathbb{R}\) and \(\varepsilon > 0\) such that \[\bigl|\delta_{p+k} - (1-\beta)\,d^* - \beta\,\delta_p\bigr| < \varepsilon \qquad\text{for all } p \in k\mathbb{N}_0 .\] Then \(\sigma^2 = 0\), \(\mathop{\mathrm{supp}}\nu \subseteq (-\infty,0]\), \(\int(|x|\wedge1)\,d\nu < \infty\), the limit \(\delta_\infty := \lim_{p\to\infty}\delta_p\) exists finitely, and \((1-\beta)\,|d^* - \delta_\infty| \le \varepsilon\). Moreover, with \(u = e^{kx}\), \(\eta = (1-u)\,d\tilde{\nu}\), and \(A = (\delta_0 - \delta_\infty)\ln r\):
(i) if \(d^* = \delta_\infty\) (the true limit is known), \[W_1\!\left(\frac{\eta}{\|\eta\|},\;\delta_\beta\right) \;\leq\; \left(\frac{(1+\beta)\,|\ln r|}{|A|}\right)^{\!1/2} \!\sqrt{\varepsilon}\,;\]
(ii) in general (fitted \(d^*\)), \[W_1\!\left(\frac{\eta}{\|\eta\|},\;\delta_\beta\right) \;\leq\; \left(\frac{2\,|\ln r|}{|A|}\right)^{\!1/2} \!\sqrt{\varepsilon}\,.\] Both constants are sharp in their respective settings. In particular, the cascade multiplier distribution converges to log-Poisson as \(\varepsilon \to 0\), at the sharp rate \(\Theta(\sqrt\varepsilon)\) (Theorems 6 and 7).
Proof. Step 0 (reduction and existence of the limit). From the hypothesis, \(|\delta_{(m+1)k}| \le \beta|\delta_{mk}| + (1-\beta)|d^*| + \varepsilon\), so the lattice sequence \(\{\delta_{mk}\}\) is bounded: \(\limsup_m |\delta_{mk}| \le |d^*| + \varepsilon/(1-\beta)\). But by Steps 2, 3 and 3½ of the proof of Theorem 3—none of which used the exact form of A1, only the finiteness of \(\liminf\) of \(\{\delta_p\}\)—each of the conditions \(\sigma^2 > 0\), \(\nu((0,\infty)) > 0\), \(\int_{|x|\le1}|x|\,d\nu = \infty\) forces \(\delta_{mk} \to -\infty\), a contradiction. Hence \(\sigma^2 = 0\), \(\mathop{\mathrm{supp}}\nu \subseteq (-\infty,0]\), \(\int(|x|\wedge1)\,d\nu < \infty\); the split form of \(\phi\) is valid, dominated convergence gives \(\phi(p) \to c_0\), and \(\delta_\infty = c_0/\ln r\) exists finitely. Letting \(m \to \infty\) in the hypothesis, \[\bigl|\delta_\infty - (1-\beta)d^* - \beta\delta_\infty\bigr| \le \varepsilon \quad\Longrightarrow\quad (1-\beta)\,|d^* - \delta_\infty| \le \varepsilon .\]
Step 1 (exact moment identities and residuals). As in Step 4 of Theorem 3 (now with no approximation in the identity itself), \[\mu_m := \int_{(0,1)} u^m\,d\eta \;=\; \bigl(\delta_{mk} - \delta_\infty\bigr)\,|\ln r| \;\ge\; 0 \qquad\text{for all } m \ge 0,\] in particular \(\|\eta\| = \mu_0 = (\delta_0 - \delta_\infty)|\ln r| = |A|\) exactly. Define the signed residuals \[\epsilon_m := \delta_{(m+1)k} - (1-\beta)\,\delta_\infty - \beta\,\delta_{mk} \;=\; \frac{\mu_{m+1} - \beta\,\mu_m}{|\ln r|},\] and let \(h_m\) denote the hypothesis residuals (with \(d^*\) in place of \(\delta_\infty\)), \(|h_m| < \varepsilon\). Setting \(t := (1-\beta)(d^* - \delta_\infty)\), \(|t| \le \varepsilon\) by Step 0, one has \(\epsilon_m = h_m + t\). Under reading (i), \(t = 0\) and \(|\epsilon_m| < \varepsilon\).
Step 2 (variance identity). Telescoping, \[\begin{align} \int_{(0,1)} (u-\beta)^2\,d\eta &= \mu_2 - 2\beta\mu_1 + \beta^2\mu_0 = (\mu_2 - \beta\mu_1) - \beta(\mu_1 - \beta\mu_0)\\ &= |\ln r|\,\bigl(\epsilon_1 - \beta\,\epsilon_0\bigr). \end{align}\] Under reading (i): \(|\epsilon_1 - \beta\epsilon_0| < (1+\beta) \varepsilon\). Under reading (ii): \(\epsilon_1 - \beta\epsilon_0 = (h_1 - \beta h_0) + (1-\beta)t\), so \(|\epsilon_1 - \beta\epsilon_0| < (1+\beta)\varepsilon + (1-\beta)\varepsilon = 2\varepsilon\). Hence \[0 \;\le\; \int_{(0,1)}(u-\beta)^2\,d\eta \;\le\; \begin{cases} (1+\beta)\,|\ln r|\,\varepsilon & \text{(i)},\\[2pt] 2\,|\ln r|\,\varepsilon & \text{(ii)}. \end{cases}\]
Step 3 (Wasserstein bound). For a Dirac target the \(W_1\) distance is the first absolute moment: \(W_1(\eta/\|\eta\|, \delta_\beta) = \|\eta\|^{-1}\int|u-\beta| \,d\eta\). By Cauchy–Schwarz and \(\|\eta\| = |A|\), \[\begin{align} W_1\!\left(\frac{\eta}{\|\eta\|},\,\delta_\beta\right) &\;\le\; \Bigl(\frac{1}{|A|}\int (u-\beta)^2\,d\eta\Bigr)^{1/2}\\ &\;\le\; \begin{cases} \bigl((1+\beta)\,|\ln r|/|A|\bigr)^{1/2}\sqrt{\varepsilon} & \text{(i)},\\[2pt] \bigl(2\,|\ln r|/|A|\bigr)^{1/2}\sqrt{\varepsilon} & \text{(ii)}. \end{cases} \end{align} \qedhere\] ◻
Remark 7 (Sharpness; a warning about per-moment transfer). (a) The constant in reading (i) is attained in the limit by the two-atom family \(\eta = c_0\delta_{u_0} + c_1\delta_v\) with \(u_0 \downarrow 0\), \(v \in (\beta,1)\), \(c_1 v(v-\beta) = \varepsilon|\ln r|\) and \(c_0 = (\varepsilon|\ln r|/\beta)(1 + 1/v)\): then \(\epsilon_0 \to -\varepsilon\), \(\epsilon_1 = +\varepsilon\), \(|\epsilon_m| = \varepsilon v^{m-1} \le \varepsilon\) for \(m \ge 2\), and \(\int(u-\beta)^2 d\eta \to (1+\beta)|\ln r|\varepsilon\).
(b) The constant in reading (ii) is likewise attained: choose moment residuals \((\epsilon_0, \epsilon_1) \to (0, 2\varepsilon)\) realized by two atoms (one near \(0\), one in \((\beta,1)\)) and the offset \(t \to \varepsilon\); all hypothesis residuals \(h_m = \epsilon_m - t\) then stay below \(\varepsilon\) in absolute value. For \(\beta < \sqrt2 - 1\) one has \(2 > (1+\beta)^2\): the distinction between the two readings is quantitatively real, and a constant valid in reading (i)—even the non-sharp \((1+\beta)^2\)—can fail* outright in reading (ii).*
(c) A tempting route passes through the per-moment estimate \(|\mu_m - |A|\beta^m| \le |\ln r|\,\varepsilon\) for all \(m\). That estimate is false in general: recurrence errors accumulate to \(|\mu_m - |A|\beta^m| \le |\ln r|\,\varepsilon\, \tfrac{1-\beta^m}{1-\beta}\), and the bound is attained in the limit by \(\eta = (|A|-c)\delta_\beta + c\,\delta_v\) with \(c(v-\beta) = \varepsilon|\ln r|\), \(v \to 1\), for which \(\sup_m |\mu_m - |A|\beta^m| / (|\ln r|\varepsilon) \to 1/(1-\beta)\). It is also unnecessary: only \(\epsilon_0\) and \(\epsilon_1\) enter the variance identity.
We now transfer the bound of Theorem 5 from the Lévy measure to the multiplier distribution, at a rate that is exact: \(O(\sqrt\varepsilon)\) with no further hypotheses (Theorem 6), and no better (Theorem 7).
Throughout this section the hypotheses are those of Theorem 5, reading (i) (for reading (ii) replace \((1+\beta)\) by \(2\) in every constant), so that Step 0 there gives \(\sigma^2 = 0\), \(\mathop{\mathrm{supp}}\nu \subseteq (-\infty,0]\), \(\int(|x|\wedge1)\,d\nu < \infty\), and Steps 2–3 give, with \(V_\varepsilon := (1+\beta)|\ln r|\,\varepsilon\) and \(S := \bigl((1+\beta)\,|A|\,|\ln r|\bigr)^{1/2}\), \[\label{eq:VF} \int_{(0,1)}(u-\beta)^2\,d\eta \;\le\; V_\varepsilon, \qquad \int_{(0,1)}|u-\beta|\,d\eta \;\le\; S\sqrt{\varepsilon} .\tag{2}\] The Lévy measure may have infinite total mass, accumulating only at \(u = 1\) where the tilt \((1-u)\) vanishes; the multiplier is the a.s.-convergent product over the Poisson point process \(\{U_i\}\) of intensity \(\tilde{\nu}\), \[W = e^{a_\varepsilon}\prod_i U_i^{1/k}, \quad \sum_i\bigl(1 - U_i^{1/k}\bigr) \le \tfrac2k\sum_i(1-U_i) \;\text{ of finite mean } \tfrac2k\|\eta\|\] (Campbell’s formula [15]). Under \(\mathbb{E}[W]=1\) the drift is \(a_\varepsilon = \int(1-u^{1/k})\,d\tilde{\nu} < \infty\). The comparison target \(W_0\) is the log-Poisson multiplier with jump factor \(\beta^{1/k}\), rate \(\lambda = |A|/(1-\beta)\), and drift \(a_0\) fixed by the same normalization.
Theorem 6 (Unconditional propagation). Under the hypotheses of Theorem 5 alone—no finite-activity or minimum-jump assumption—there exist \(\varepsilon_0 > 0\) and an explicit constant \(K_\infty = K_\infty(\beta, C, r, k)\) such that for all \(\varepsilon \le \varepsilon_0\), \[W_1\bigl(\mathrm{law}(W),\;\mathrm{law}(W_0)\bigr) \;\le\; K_\infty\,\sqrt{\varepsilon}.\] One admissible (not optimized) choice is \[\begin{align} K_\infty &= 4\,e^{a_0+1}\Bigl(\frac{L_k}{1-\beta} + \frac{1}{(1-\beta)^2}\Bigr) \sqrt{(1+\beta)\,|A|\,|\ln r|}\,,\\ L_k &= \tfrac1k\bigl(\tfrac\beta2\bigr)^{(1-k)/k}, \end{align}\] absorbing \(O(\varepsilon)\) terms via \(\varepsilon \le \sqrt\varepsilon\).
Proof. Fix the \(\varepsilon\)-independent split height \[h_0 := \tfrac{1-\beta}{2}, \qquad u_0 := 1 - h_0 = \tfrac{1+\beta}{2},\] and call a jump small if \(u \in (u_0, 1)\), macroscopic if \(u \in (0, u_0]\).
Step 1 (small jumps: individually cheap, collectively \(O(\varepsilon)\)). Every \(u \in (u_0,1)\) lies at distance \(> h_0\) from \(\beta\), so by Chebyshev’s inequality against 2 , \[\eta\bigl((u_0,1)\bigr) \;\le\; \frac{V_\varepsilon}{h_0^{\,2}} \;=\; \frac{4(1+\beta)|\ln r|}{(1-\beta)^2}\,\varepsilon .\] Leave the small-jump points of \(W\) unpaired and cost them multiplicatively: for factors in \((0,1]\), telescoping gives \(|\prod_i s_i - 1| \le \sum_i(1-s_i)\) (valid for infinite products by monotone limits), and \(1 - u^{1/k} \le \tfrac2k(1-u)\) for \(u \ge \tfrac12\) (derivative bound; \(u_0 \ge \tfrac12\)). By Campbell’s formula, \[\begin{align} \mathbb{E}\Bigl|\prod_{\text{small}} U_i^{1/k} - 1\Bigr| \;&\le\; \frac{2}{k}\int_{(u_0,1)}(1-u)\,d\tilde{\nu} \;=\; \frac{2}{k}\,\eta\bigl((u_0,1)\bigr)\\ \;&\le\; \frac{8(1+\beta)|\ln r|}{k(1-\beta)^2}\,\varepsilon . \end{align}\] This is the step that controls infinite activity: a near-1 jump’s cost is its multiplicative deviation \(\asymp(1-u)\)—already \(\eta\)-weighted—not the count \(1\).
Step 2 (the macroscopic part is automatically finite-activity, with a fixed de-tilting constant). On \((0,u_0]\): \(\tfrac1{1-u} \le \tfrac1{h_0} = \tfrac2{1-\beta}\), so \(\lambda_{\mathrm{mac}} := \tilde{\nu}((0,u_0]) \le \tfrac{2|A|}{1-\beta} < \infty\). Comparing rates against \(\lambda = |A|/(1-\beta) = \int\tfrac{d\eta}{1-\beta}\), and using \(\bigl|\tfrac1{1-u} - \tfrac1{1-\beta}\bigr| = \tfrac{|u-\beta|}{(1-u)(1-\beta)} \le \tfrac{2|u-\beta|}{(1-\beta)^2}\) on \((0,u_0]\), \[\bigl|\lambda_{\mathrm{mac}} - \lambda\bigr| \;\le\; \frac{2}{(1-\beta)^2}\,S\sqrt{\varepsilon} \;+\; \frac{\eta((u_0,1))}{1-\beta},\] the second term being \(O(\varepsilon)\) by Step 1.
Step 3 (macroscopic jump cost). The map \(u \mapsto u^{1/k}\) is \(L_k\)-Lipschitz on \([\beta/2,1]\) and bounded by 1 below \(\beta/2\); on \((0,\beta/2)\), \(\tfrac1{1-u} \le \tfrac1{1-\beta/2} \le 2\), so \(\tilde{\nu}((0,\beta/2)) \le 2\eta((0,\beta/2)) \le 2V_\varepsilon(\beta/2)^{-2}\) by Chebyshev. Hence \[\int_{(0,u_0]}\bigl|u^{1/k} - \beta^{1/k}\bigr|\,d\tilde{\nu} \;\le\; \frac{2L_k}{1-\beta}\,S\sqrt{\varepsilon} \;+\; \frac{8\,V_\varepsilon}{\beta^2}.\]
Step 4 (assembly). Couple the macroscopic Poisson count with \(W_0\)’s count by thinning (shared count \(\mathrm{Poisson}(\lambda\wedge\lambda_{\mathrm{mac}})\), excess independent; [16], Theorem 10.A), shared jump pairs optimally in the multiplier coordinate against the constant target \(\beta^{1/k}\), and leave the small jumps unpaired. Writing \(W = e^{a_\varepsilon}P_{\mathrm{mac}}P_{\mathrm{small}}\) and \(W_0 = e^{a_0}P_0\) with all products in \([0,1]\), the telescoping inequality and Wald’s identity give \[\begin{align} \mathbb{E}\bigl|P_{\mathrm{mac}}P_{\mathrm{small}} - P_0\bigr| \;\le\; &\underbrace{\int_{(0,u_0]}\!\bigl|u^{1/k}-\beta^{1/k}\bigr| d\tilde{\nu}}_{\text{Step 3}} \;+\; \underbrace{\bigl|\lambda_{\mathrm{mac}} - \lambda\bigr|}_{\text{Step 2}}\\ &+\; \underbrace{\mathbb{E}\bigl|P_{\mathrm{small}} - 1\bigr|}_{\text{Step 1}}, \end{align}\] each excess jump on either side changing a product by at most 1. The drift difference obeys the same bound: under \(\mathbb{E}[W]=1\), \(a_\varepsilon - a_0 = \int(1-u^{1/k})\,d\tilde{\nu} - \lambda(1-\beta^{1/k})\) decomposes into the same three pieces. Both multipliers are bounded by \(e^{a_\cdot}\) (Lemma 2(i)), so for \(\varepsilon \le \varepsilon_0\) with \(|a_\varepsilon - a_0| \le 1\), \[\begin{align} W_1 \le \mathbb{E}|W - W_0| &\le e^{a_0+1}\Bigl(\mathbb{E}|P_{\mathrm{mac}}P_{\mathrm{small}} - P_0| + |a_\varepsilon - a_0|\Bigr)\\ &\le 2e^{a_0+1}\bigl[\text{Step 1} + \text{Step 2} + \text{Step 3}\bigr]. \end{align}\] Substituting the three displays and absorbing every \(O(\varepsilon)\) term via \(\varepsilon \le \sqrt\varepsilon\) yields the stated \(K_\infty\). ◻
Remark 8 (Why no hypotheses are needed). A coupling built in log-space would require finite activity and a minimum jump size—a Poisson count must be finite, and a de-tilting factor must be bounded—and an \(\varepsilon\)-dependent shell decomposition for the small jumps would surrender a logarithmic factor, \(O(\sqrt\varepsilon\log(1/\varepsilon))\). In the multiplicative coupling both holes close themselves: infinite activity can only accumulate at \(u = 1\), where multiplicative cost vanishes at exactly the rate \(\eta\) measures, and the de-tilting factor appears only on \((0,u_0]\), where it is the fixed constant \(2/(1-\beta)\). The flatness of the exponential map at \(-\infty\), fatal to couplings of jump distributions in \(x\)-space, is the resource here.
Theorem 7 (Lower bound: the rate \(\sqrt\varepsilon\) is exact). Fix \((\beta, C, r)\), \(k = 1\), and conservation \(\mathbb{E}[W]=1\). For \(d \in (0, d_0]\), \(d_0 = d_0(\beta)\) small, let \(W_{(d)}\) be the compound-Poisson multiplier with tilted measure \[\eta_d = \tfrac{|A|}{2}\bigl(\delta_{\beta-d} + \delta_{\beta+d}\bigr)\] (jump atoms \(\beta\pm d\) with Lévy masses \(\tfrac{|A|/2}{1-(\beta\pm d)}\), drift by conservation). Then:
(i) its A1 residual satisfies \(\dfrac{|A|}{|\ln r|}\,d^2 \;\le\; \varepsilon(d) \;\le\; c_2(\beta)\,\dfrac{|A|}{|\ln r|}\,d^2\), where \(c_2(\beta) = \bigl[e\min_{\rho\in[\beta,(1+\beta)/2]}\rho|\ln\rho|\bigr]^{-1} \vee 1\);
(ii) with \(\lambda_d = \|\tilde{\nu}_d\| = \lambda/(1 - d^2(1-\beta)^{-2})\) and \(a = |A|\), \[W_1\bigl(\mathrm{law}\,W_{(d)},\;\mathrm{law}\,W_0\bigr) \;\ge\; \lambda_d\,e^{-\lambda_d}\,e^{a}\,d \;\ge\; c_1(\beta, C, r)\,\sqrt{\varepsilon(d)}\,.\] Hence no propagation bound of order \(o(\sqrt\varepsilon)\) is possible: combined with Theorem 6, the exact rate is \(\Theta(\sqrt\varepsilon)\).
Proof. (i) The moments are \(\mu_m = \tfrac{|A|}{2}[(\beta-d)^m + (\beta+d)^m]\), so the signed residuals are \[\epsilon_m = \frac{\mu_{m+1} - \beta\mu_m}{|\ln r|} = \frac{|A|\,d}{2\,|\ln r|} \bigl[(\beta+d)^m - (\beta-d)^m\bigr] \;\ge 0,\] with \(\epsilon_0 = 0\) and \(\epsilon_1 = |A|d^2/|\ln r|\) exactly, giving the lower bound on \(\varepsilon(d) = \sup_m \epsilon_m\). For the upper bound, the mean value theorem gives \((\beta+d)^m - (\beta-d)^m \le 2dm(\beta+d)^{m-1}\), and \(\sup_{x\ge0} x\rho^{x-1} = (e\rho|\ln\rho|)^{-1}\) for \(\rho = \beta+d \le (1+\beta)/2\).
(ii) For \(k=1\) the conservation drift is \(a_{(d)} = \int(1-u)\,d\tilde{\nu}_d = \|\eta_d\| = |A|\), identical to \(a_0 = \lambda(1-\beta) = |A|\): the family is drift-rigid. The support of \(\mathrm{law}(W_0)\) is the geometric set \(G = \{e^{a}\beta^j : j \ge 0\}\). Take the 1-Lipschitz test function \[f(w) := \min\Bigl(\mathop{\mathrm{dist}}\bigl(w,\,G\bigr),\; \tfrac{e^a\beta(1-\beta)}{2}\Bigr) \;\ge\; 0,\] which vanishes on \(G\), so \(\mathbb{E}f(W_0) = 0\). On the one-jump event of \(W_{(d)}\) (probability \(\lambda_d e^{-\lambda_d}\)), \(W_{(d)} = e^{a}(\beta\pm d)\), whose distance to the nearest point of \(G\) is exactly \(e^a d\) for \(d < \beta(1-\beta)/2\) (the neighbors \(e^a\) and \(e^a\beta^2\) are farther, and \(e^ad\) is below the cap); all other events contribute \(\ge 0\). By Kantorovich–Rubinstein duality [17], \[W_1 \;\ge\; \mathbb{E}f(W_{(d)}) - \mathbb{E}f(W_0) \;\ge\; \lambda_d e^{-\lambda_d}\,e^{a}\,d,\] and substituting \(d \ge \bigl(|\ln r|\,\varepsilon(d) / (c_2(\beta) |A|)\bigr)^{1/2}\) from (i) gives the \(c_1\sqrt{\varepsilon}\) form. (The same construction works for \(k > 1\) on the lattice \(\{e^{a}\beta^{j/k}\}\); \(k = 1\) is stated for cleanliness.) ◻
Remark 9 (The stability theory is elementary and now complete). The change of variables \(u = e^{kx}\) maps the Lévy measure to the compact interval \([0,1]\), where a second-moment test against the candidate atom decides everything: the classification is the \(\varepsilon = 0\) case of the variance identity in Theorem 5, Step 2; the stability constant is read off two residuals and is sharp; and the propagation to the multiplier law is a telescoped multiplicative coupling with no tail estimates (boundedness, Lemma 2(i)), sharp in rate by Theorem 7. Every quantitative statement in the package—variance constant \((1+\beta)\), Wasserstein constant, propagation rate \(\Theta(\sqrt\varepsilon)\)—is attained by an explicit family. In particular the log-Poisson class is an open set, with exactly computed modulus, in the space of cascade multiplier distributions metrized by A1 residuals.
The i.i.d.assumption enters the preceding sections through the identity \(\mathbb{E}[(W_1\cdots W_n)^p] = (\mathbb{E}[W^p])^n\), which converts scaling data into one-step moment data. This section determines exactly what survives without it. Let \((W_n)\) be stationary ergodic, \(X_n = \ln W_n\), \(S_n = X_1 + \cdots + X_n\), and define exponents asymptotically:
Standing assumptions (S). For each lattice \(p\): \(m_n(p) := \mathbb{E}[e^{pS_n}] < \infty\) for all \(n\), and \[\Lambda(p) := \lim_{n\to\infty}\tfrac1n\ln m_n(p) \quad\text{exists and is finite}; \qquad \zeta_p := \Lambda(p)/\ln r .\] A1 is imposed on \(\delta_p = \zeta_{p+k}-\zeta_p\) as before. Write \(\Lambda_{\mathrm{LP}}(p) = ap + \lambda(e^{bp}-1)\) for the log-Poisson limiting cumulant function with the parameter dictionary of Theorem 1.
Theorem 8 (Asymptotic-statistics classification). Under (S), A1 holds on the lattice if and only if \(\Lambda = \Lambda_{\mathrm{LP}}\) on the lattice. Consequently, under A1 every observable computed from lattice exponents—structure-function exponents, moment ratios, and (under the regularity below) the large-deviation rate function and multifractal spectrum—coincides exactly with that of the i.i.d.log-Poisson cascade with parameters \((\beta, \gamma, C)\).
Proof. Lemma 1 is purely algebraic, so A1 gives \(\Lambda(km) = a\,km + \lambda(e^{b\,km}-1)\) for all \(m\); the converse is the computation of Proposition 2. ◻
Remark 10. If \(\Lambda\) exists, is finite and differentiable on a neighborhood of \([0,\infty)\), the Gärtner–Ellis theorem [18] yields a large-deviation principle for \(S_n/n\) with rate \(I(h) = \sup_p[ph - \Lambda(p)]\) on the exposed range—the Legendre structure of Theorem 1(iii). Off-lattice, \(\Lambda\) is pinned between consecutive lattice values by convexity.
Theorem 8 is deliberately easy; the substantive question is whether A1 still determines the multiplier law. It does not:
Theorem 9 (Interleaved cascade: the law is not determined). There exists a stationary ergodic multiplier sequence \((W_n)\)—realizable as a function of a stationary, irreducible, positive-recurrent countable-state Markov chain—such that:
(i) \(m_n(p) = r^{n\zeta^{\mathrm{LP}}_p}\) exactly* for every even \(n\) and every real \(p \ge 0\); hence (S) holds, \(\Lambda = \Lambda_{\mathrm{LP}}\) on all of \([0,\infty)\), and A1 holds exactly, in its strongest (real-\(p\)) form;*
(ii) the one-step marginal of \(\ln W_1\) is not log-Poisson;
(iii) the sequence is not i.i.d., and is not equal in law to any i.i.d.cascade.
Consequently the law-level conclusion of Theorem 1 does not extend beyond independence, by any proof.
Proof. Construction. With \((a, b, \lambda)\) as in Theorem 1 and \(\lambda' := 2\lambda\), let \(A_1, A_2, \ldots\) be i.i.d.\(\mathrm{Poisson}(\lambda')\), define \[Y_{2j-1} = a + bA_j, \qquad Y_{2j} = a \qquad (j \ge 1),\] draw a phase \(\theta \sim \mathrm{Unif}\{0,1\}\) independent of everything, and set \(X_n := Y_{n+\theta}\), \(W_n := e^{X_n}\): jump slots of doubled intensity alternate with deterministic slots.
Stationarity and Markov realization. The pair \(Z_n := ((n+\theta)\bmod 2,\, X_n)\) is a Markov chain on \(\{0,1\}\times(a+b\mathbb{N}_0)\): from phase-1 states the next value is \(a + bA\) with fresh \(A \sim \mathrm{Poisson}(\lambda')\) and the phase flips; from phase-0 states the next value is \(a\) and the phase flips. The chain is irreducible on its reachable set and positive recurrent; the phase-uniform stationary law makes \((Z_n)\) stationary and \(W_n\) a function of it.
Ergodicity. Let \(P_0, P_1\) be the path laws given \(\theta = 0,1\), so the law is \(\tfrac12(P_0+P_1)\) with \(P_1 = P_0\circ T^{-1}\) (\(T\) = shift). If \(E\) is \(T\)-invariant, it is \(T^2\)-invariant, and under \(P_0\) the double shift is ergodic (the blocks \((Y_{2j-1}, Y_{2j})\) are i.i.d.), so \(P_0(E) \in \{0,1\}\); and \(P_1(E) = P_0(T^{-1}E) = P_0(E)\). Hence \(\mathbb{P}(E) \in \{0,1\}\).
Exact exponents. For even \(n = 2m\) the window \(\{1,\ldots,2m\}\) contains exactly \(m\) jump slots under either phase, carrying \(m\) distinct i.i.d.\(A_j\)’s; hence, exactly, for every real \(p\), \[\begin{align} m_{2m}(p) &= e^{2map}\exp\bigl[m\lambda'(e^{bp}-1)\bigr] = \exp\bigl[2m\bigl(ap + \lambda(e^{bp}-1)\bigr)\bigr]\\ &= \exp\bigl[2m\,\Lambda_{\mathrm{LP}}(p)\bigr]. \end{align}\] For odd \(n\) the window covers \(m\) or \(m+1\) jump slots depending on the phase, and \(\tfrac1n\ln m_n(p) \to \Lambda_{\mathrm{LP}}(p)\). Theorem 8 then gives A1 exactly.
Non-log-Poisson marginal. \(\mathbb{P}(X_1 = a) = \tfrac12(1 + e^{-2\lambda})\), whereas the log-Poisson(\(\lambda\)) marginal has \(\mathbb{P}(X = a) = e^{-\lambda}\); for She–Lévêque dissipation parameters (\(\lambda = 2\ln2\)): \(0.531\) versus \(0.250\). The marginal is the mixture \(\tfrac12\delta_a + \tfrac12\,\mathrm{law}(a + b\,\mathrm{Poisson}(2\lambda))\).
Non-i.i.d. Given \(X_n \ne a\) the next multiplier is deterministic: \(\mathbb{P}(X_{n+1} = a \mid X_n \ne a) = 1 \ne \mathbb{P}(X_{n+1} = a)\). ◻
Remark 11. The construction generalizes freely (blocks of length \(L\), arbitrary allocation of the total jump intensity across slots, Markov-modulated loads): A1 is compatible with an infinite-dimensional family of mutually singular stationary ergodic processes, all sharing the log-Poisson asymptotics—exactly as Theorem 8 says they must.
Two rigidity results delimit the boundary of Theorem 9.
Corollary 4 (Exchangeable rigidity). Let \((W_n)\) be exchangeable with \(m_n(p) = r^{n\zeta_p}\) holding exactly for \(n \in \{1,2\}\) and all lattice \(p\), with \(\zeta\) satisfying A1. Then \((W_n)\) is i.i.d.log-Poisson with parameters \((\beta, \gamma, C)\).
Proof. By de Finetti’s theorem [19], \((W_n)\) is conditionally i.i.d.given a random directing measure; let \(M_p := \mathbb{E}[W_1^p \mid \text{directing measure}]\). Conditional independence gives \(m_1(p) = \mathbb{E}[M_p]\) and \(m_2(p) = \mathbb{E}[M_p^2]\), so exactness at \(n = 1,2\) reads \(\mathbb{E}[M_p] = r^{\zeta_p}\), \(\mathbb{E}[M_p^2] = r^{2\zeta_p}\), whence \(\mathop{\mathrm{Var}}(M_p) = 0\): \(M_p = r^{\zeta_p}\) a.s., for every lattice \(p\) simultaneously. Almost every directing measure therefore has exactly the A1 lattice moments, hence equals the log-Poisson law by Lemma 2 and Theorem 1(ii); the mixture is degenerate. ◻
Theorem 10 (Finite-state impossibility). Let \(\xi\) be an irreducible finite-state Markov chain, stationary, and \(W_n = e^{f(\xi_n)}\) with \(f\) non-constant. Then the cascade cannot satisfy A1 in its real-\(p\) form with nontrivial intermittency: there are no parameters \((\beta, \gamma, C)\) with \(C > 0\) and \(\zeta_p = \gamma p + C(1-\beta^{p/k})\) for all real \(p \ge 0\).
Proof. For finite irreducible chains, \(\Lambda(p) = \ln\rho(M(p))\) with \(M(p)_{xy} = P_{xy}e^{pf(y)}\) and \(\rho\) the Perron root—a simple eigenvalue for every real \(p\), hence real-analytic on \(\mathbb{R}\). If the closed form held on \([0,\infty)\), then \(\Lambda(p) = ap + \lambda(e^{bp}-1)\) there with \(\lambda = -C\ln r > 0\); both sides are real-analytic on \(\mathbb{R}\) and agree on an interval, hence agree everywhere (identity theorem). Now let \(p \to -\infty\). Since \(f_{\min} = \min_s f(s) > -\infty\), every row sum of \(M(p)\) is at most \(e^{pf_{\min}}\) for \(p \le 0\), so \(\Lambda(p) \le pf_{\min}\): a linear upper bound. But with \(b < 0\), \[\Lambda_{\mathrm{LP}}(p) - pf_{\min} \;\ge\; \lambda e^{|b||p|} - \lambda + p(a - f_{\min}) \;\longrightarrow\; +\infty :\] contradiction. ◻
Remark 12 (The boundary, and what A1 really is). The mechanism of Theorem 10 is that finite alphabets make \(\ln W\) bounded below, while the log-Poisson generator is intrinsically unbounded below (\(\ln W \in a + b\mathbb{N}_0\)): A1 forces multipliers with arbitrarily severe attenuation events, and the counterexample of Theorem 9 necessarily has unbounded-below \(\ln W\). Assembled, this section says: A1 is an asymptotic-statistics* axiom. It pins the limiting cumulant function, the spectrum, and the large deviations to the log-Poisson cascade (Theorem 8); it cannot pin the per-step law (Theorem 9); and the i.i.d.log-Poisson cascade is the canonical realization—unique under exchangeability (Corollary 4), with finite-state Markov realizations impossible (Theorem 10). Whether Theorem 10 persists under lattice-only A1 remains open (the identity-theorem step is unavailable on a discrete set; see Section 9).*
We now place the theory in the continuous category of Bacry–Muzy multifractal random measures [5], [11], which contains the log-normal multifractal random walk, log-stable measures, and the Barral–Mandelbrot compound Poisson cascades [4] as special cases. We use two structural properties of the class as axioms:
(M1) Exact stochastic scale invariance. For every \(\sigma \in (0,1)\) and \(t \le T\) (the integral scale), \[\bigl(M(\sigma t)\bigr)_t \;\stackrel{d}{=}\; \sigma\,e^{\Omega_\sigma}\bigl(M(t)\bigr)_t,\] with \(\Omega_\sigma\) infinitely divisible, independent of \(M\), and \(\mathbb{E}[e^{q\Omega_\sigma}] = \sigma^{-\psi(q)}\), where \(\psi\) is the Lévy exponent of the generator per unit logarithmic scale.
(M2) Conservation. \(\mathbb{E}[e^{\Omega_\sigma}] = 1\), i.e.\(\psi(1) = 0\).
(S\(_c\)) \(\psi(q) < \infty\) for all \(q \ge 0\) (the continuous analogue of finite multiplier moments).
From (M1), wherever \(\mathbb{E}[M([0,t])^q] < \infty\), \[\mathbb{E}\bigl[M([0,t])^q\bigr] \propto t^{\zeta_q}, \qquad \zeta_q = q - \psi(q),\] and moments of the total mass are finite (for \(q > 1\)) precisely on the window where \(\zeta_q > 1\) [5]. Since \(\zeta\) is concave with \(\zeta_1 = 1\), structure functions see only a finite window \([0, q^*)\)—a hard information barrier with no discrete counterpart (there the log-Poisson multiplier is bounded and all moments exist). The hierarchical symmetry A1 is imposed on \(\delta_q = \zeta_{q+k} - \zeta_q\) as before.
The dictionary to the discrete theory is one line: with \(\phi_c(q) := \psi(q+k) - \psi(q)\) one has \(\delta_q = k - \phi_c(q)\), and A1 gives, by Lemma 1, \[\phi_c(mk) - \phi_{c,\infty} = A_c\,\beta^m, \qquad A_c = -(\delta_0 - \delta_\infty) = -C(1-\beta) < 0 :\] formally the discrete identities with \(\ln r \mapsto -1\), so \(|A_c| = C(1-\beta)\) and the rate \(\lambda = -C\ln r\) becomes the intensity \(C\) per unit log-scale. Every \(|\ln r|\) in the discrete constants disappears.
Theorem 11 (No finite moment window identifies the class). Fix the She–Lévêque exponent curve \(\zeta^{\mathrm{SL}}_q = \gamma q + C(1-\beta^{q/k})\) normalized by (M2), and any finite lattice window \(F \subset k\mathbb{N}_0\). Then there is a continuum of exactly scale-invariant log-ID multifractal random measures whose generators are not compound Poisson with a single atom—in particular are not the compound Poisson cascade—yet whose exponents satisfy \(\zeta_q = \zeta^{\mathrm{SL}}_q\) for every \(q \in F\). Structure-function data on a finite moment window, even exact and noise-free, cannot certify the log-Poisson class.
Proof. We give the construction for \(k = 1\) and the physically typical window \(F = \{0,1,2\}\) (i.e.\(2 < q^* \le 3\)); larger windows are identical with more atoms. In the tilted coordinates \(u = e^{x}\), the CPC generator (Theorem 12) has \(\tilde{\Pi} = C \delta_\beta\). Perturb: \[\tilde{\Pi}_s = (C - sc)\,\delta_\beta + s\,(w_1\delta_{v_1} + w_2\delta_{v_2}), \qquad 0 < v_1 < \beta < v_2 < 1,\] with the drift re-fixed by (M2) for each \(s\), and impose \[\sum_{i=1,2} w_i\,(v_i^m - 1) \;=\; c\,(\beta^m - 1), \qquad m = 1, 2 .\] The \(m=1\) equation makes the (M2)-drifts of \(\tilde{\Pi}_s\) and \(\tilde{\Pi}_0\) coincide; the \(m=2\) equation then matches \(\zeta_2\); \(\zeta_0 = 0\) and \(\zeta_1 = 1\) are automatic. At \((\beta, v_1, v_2) = (2/3,\,0.3,\,0.9)\), \(c = 1\), the \(2\times2\) system gives \(w_1 = 0.1852\), \(w_2 = 2.0370\), both positive, so \(\tilde{\Pi}_s \ge 0\) for all \(s \in [0, C/c]\): a one-parameter family of genuine Lévy measures, none a single atom for \(s > 0\), all matching \(\zeta^{\mathrm{SL}}\) exactly on \(F\) (and differing beyond: at \(m = 3\) the perturbed exponent differs by \(+0.0143\) at \(s = \tfrac12\)). For general finite \(F\), the same ansatz with more atoms imposes finitely many linear constraints on infinitely many degrees of freedom, with positivity maintained by anchoring the negative part on the CPC atom. ◻
Remark 13. Theorem 11 does not contradict the discrete classification: there A1 was available at all* lattice orders—the divergence steps of Theorem 3 need \(q \to \infty\), and on a finite window even \(\sigma_0^2 > 0\) survives (a quadratic log-normal exponent interpolates any three-point window with the correct convexity). Identifiability requires constraints of unbounded order, which structure functions cannot supply. The scale-invariance factor \(\Omega_\sigma\) can: under (S\(_c\)) it has all exponential moments, at every \(\sigma\), and its statistics (“magnitude” statistics, in the language of [12]) are observable at all orders.*
Theorem 12 (Generator-level classification: A1 selects the compound Poisson cascade). Let \(M\) satisfy (M1), (M2), (S\(_c\)), with nontrivial intermittency \(C > 0\), and define the generator exponents \(\zeta_q = q - \psi(q)\) for all \(q \ge 0\). Then A1 on the full lattice \(k\mathbb{N}_0\) holds if and only if \[\sigma_0^2 = 0, \qquad \Pi = C\,\delta_b, \qquad b = \frac{\ln\beta}{k} < 0,\] with drift \(\tilde{a} = C(1-\beta^{1/k})\) fixed by (M2); equivalently \[\Omega_\sigma \;\stackrel{d}{=}\; \tilde{a}\,\ln(1/\sigma) \;+\; b\,\mathrm{Poisson}\bigl(C \ln(1/\sigma)\bigr) \qquad\text{for every } \sigma \in (0,1),\] log-Poisson at every scale ratio simultaneously, and \(M\) is the Barral–Mandelbrot compound Poisson cascade [4] with fixed multiplier atom \(e^b = \beta^{1/k}\) and Poisson intensity \(C\) on the time–log-scale cone. No other member of the Bacry–Muzy class—log-normal, log-stable, or any intermediate—satisfies A1.
Proof. Reverse: with \(\Pi = C\delta_b\), \(\psi(q) = \tilde{a}q + C(e^{bq}-1)\), so \(\zeta_q = q - \psi(q) = (1-\tilde{a})q + C(1-\beta^{q/k})\), which satisfies A1 by Proposition 2 with \(\gamma = 1 - \tilde{a}\); (M2) gives \(\tilde{a} = C(1-\beta^{1/k})\), the continuous conservation constraint (\(\zeta_1 = 1\)).
Forward: \(\psi\) is a Lévy–Khintchine exponent, finite for all \(q \ge 0\), and \(\phi_c(q) = \psi(q+k)-\psi(q)\) has a finite limit under A1: exactly the hypothesis configuration of Theorem 3’s proof under the dictionary \(\ln r \mapsto -1\). Steps 2, 3, 3½ and 4 of that proof apply verbatim: a Gaussian component, positive jumps, or \(\int_{|x|\le1}|x|\,d\Pi = \infty\) each force \(\delta_q = k - \phi_c(q) \to -\infty\); then with \(\eta = (1-u)\,d\tilde{\Pi}\), \(u = e^{kx}\), A1 gives \(\int u^m\,d\eta = |A_c|\beta^m\) and the variance identity yields \(\eta = |A_c|\delta_\beta\), i.e. \(\tilde{\Pi} = \frac{|A_c|}{1-\beta}\delta_\beta = C\delta_\beta\) and \(\Pi = C\delta_b\). The identification of the compound-Poisson member of the Bacry–Muzy class with the Barral–Mandelbrot cascade is [5]. ◻
Corollary 5 (Stability with native constants). In the setting of Theorem 12, if A1 holds to within \(\varepsilon\) on the generator lattice—with \(d^*\) the true limit (reading (i)) or fitted (reading (ii))—then \(\sigma_0^2 = 0\), \(\Pi\) is supported on \((-\infty,0]\), and with \(\eta = (1-u)\,d\tilde{\Pi}\), \(\|\eta\| = |A_c| = C(1-\beta)\) exactly: \[W_1\!\Bigl(\frac{\eta}{\|\eta\|},\,\delta_\beta\Bigr) \;\le\; \sqrt{\frac{1+\beta}{C(1-\beta)}}\;\sqrt{\varepsilon} \quad\mathrm{(i)}, \qquad \le\; \sqrt{\frac{2}{C(1-\beta)}}\;\sqrt{\varepsilon} \quad\mathrm{(ii)},\] both sharp; for She–Lévêque values \((\beta, C) = (2/3, 2)\): \(1.58\sqrt\varepsilon\) and \(1.73\sqrt\varepsilon\). The propagation to the law of the per-octave factor \(e^{\Omega_{1/2}}\) holds at the unconditional sharp rate \(\Theta(\sqrt\varepsilon)\) of Theorems 6–7 (whose proofs nowhere used finite activity—the natural situation here).
Proof. Theorem 5, Steps 0–3, under the dictionary \(\ln r \mapsto -1\), \(|A| \mapsto |A_c| = C(1-\beta)\); the sharpness families transfer symbol-for-symbol, as does the propagation argument of Section 5. ◻
Remark 14 (What the two theorems mean together). Theorems 11 and 12 are two halves of one statement about observability: structure functions cannot identify the cascade class even in principle (a finiteness barrier, not a statistical one), while magnitude statistics classify it completely, with quantitative stability. This places a theorem under the long-standing practical preference for magnitude-cumulant analysis over high-order structure functions [12]. Note also that the continuous category is more* rigid than the discrete one: infinite divisibility, an assumption in Theorem 3, is automatic here (consistency of \(\Omega_{\sigma\sigma'} \stackrel{d}{=} \Omega_\sigma + \Omega'_{\sigma'}\)), so the classification needs no distributional hypothesis beyond membership in the class. Finally, the most-singular-branch geometry (Corollary 1) reads natively: the probability that the cone above a point carries no Poisson point down to scale \(\ell\) is \(\ell^{\,C}\)—codimension \(C\), with no discretization anywhere.*
Corollary 6 (Conservation constraint). If there exists an index \(k_0 > 0\) such that \(\zeta_{k_0} = z_0\) for a known constant \(z_0\) fixed by an exact conservation law, then \[\gamma = \frac{z_0 - C(1-\beta^{k_0/k})}{k_0}.\] This reduces the observable parameters from two to one.
Corollary 7 (Codimension identification). If the most singular structures have Hausdorff codimension \(C_{\mathrm{geom}}\) and \(C = C_{\mathrm{geom}}\), then \(\beta\) alone determines the full exponent curve, the multifractal spectrum, and the cascade distribution.
Corollary 8 (Spectrum width). The width of the multifractal spectrum is \[\Delta h = h_{\max} - h_{\min} = \frac{C}{k}\,|\ln\beta|,\] where \(h_{\max} = \gamma + (C/k)|\ln\beta|\) (at \(p = 0\), most regular) and \(h_{\min} = \gamma\) (at \(p \to \infty\), most singular).
Corollary 9 (Parameter-free stability constant). Since \(|A| = C(1-\beta)\,|\ln r|\), the scale ratio cancels in Theorem 5: \[W_1\!\left(\frac{\eta}{\|\eta\|},\,\delta_\beta\right) \;\le\; \sqrt{\frac{1+\beta}{C(1-\beta)}}\;\sqrt{\varepsilon} \quad\mathrm{(i)}, \qquad \le\; \sqrt{\frac{2}{C(1-\beta)}}\;\sqrt{\varepsilon} \quad\mathrm{(ii)},\] independent of \(r\) and \(k\) (and identical to the native continuous constants of Corollary 5). For fully developed turbulence (\(\beta = 2/3\), \(C = 2\), in either the dissipation form \(k=1\) or the velocity form \(k=3\)) the constants are \(\sqrt{5/2} \approx 1.58\) and \(\sqrt{3} \approx 1.73\): a measured violation \(\varepsilon\) of hierarchical symmetry confines the normalized tilted jump measure within \(1.58\sqrt{\varepsilon}\) (resp.\(1.73\sqrt{\varepsilon}\)) of \(\delta_{2/3}\) in Wasserstein-1 distance.
The results of this paper show that the hierarchical symmetry A1 carries considerably more force than might be expected from its appearance as a simple linear recurrence. Within i.i.d. multiplicative cascades it is equivalent to the log-Poisson class, with sharp stability constants and the exact propagation rate \(\Theta(\sqrt\varepsilon)\); beyond independence it pins all asymptotic statistics (and provably nothing more); and in the continuous category it selects exactly the compound Poisson cascade at the generator level, while no finite moment window of structure functions can do so. The following directions remain open.
Lattice-only finite-state rigidity. Theorem 10 assumes the closed exponent form for all real \(p \ge 0\); under lattice-only A1 the identity-theorem step is unavailable, and Carlson-type interpolation is blocked by possible complex eigenvalue crossings of the tilted transfer matrix. We expect the conclusion to persist.
Determination of \(k\), and the joint-in-\(k\) test. The hierarchy step \(k\) is treated as given; in applications it must be estimated. A1 at several steps simultaneously imposes the compatibility constraint \(\ln\beta(k) \propto k\), and the statistical gain from the joint test is unquantified.
Boundary cases. The log-normal class is the \(\beta \to 1\) closure point of the log-Poisson family (\(b \to 0\), \(\lambda b^2 \to \sigma^2\)): quantifying the degeneration of identifiability as \(\beta \uparrow 1\) would unify the classification with its principal rival. The maximal-intermittency limit \(\beta \to 0\) likewise deserves analysis.
Statistics of the A1 test. The present results are exact-population statements. A finite-sample theory—error bars on \((\hat{\beta}, \hat{d}^*)\), power against log-normal and log-stable alternatives, with the reading-(ii) constants of Theorem 5 and Corollary 5 as the operative null band—is the missing link between the classification and data; the window obstruction of Theorem 11 dictates that such a theory be built on magnitude statistics rather than high-order structure functions.
The author received no funding for this work.
The author declares no competing interests.
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