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Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability

E. M. Freeburg

Abstract

Within i.i.d. multiplicative cascades, a single axiom -- the hierarchical symmetry, a linear contraction on incremental scaling exponents -- is shown to be necessary and sufficient for the cascade multiplier to be log-Poisson. We establish three results: (1) a characterization theorem proving that the hierarchical symmetry uniquely determines the log-Poisson distribution with explicit parameters; (2) a classification theorem proving that the hierarchical symmetry selects exactly the log-Poisson class from the full log-infinitely-divisible family, excluding log-normal, log-stable, and all intermediate generators; and (3) a stability theorem proving that approximate hierarchical symmetry implies approximate log-Poisson, with an explicit $O(\sqrt{\varepsilon})$ Wasserstein bound. The proofs reduce the problem to the Hausdorff moment problem on $[0,1]$ via the change of variables $u = e^{kx}$, where determinacy and stability follow from classical results.

Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability

Abstract

Within i.i.d. multiplicative cascades, a single axiom -- the hierarchical symmetry, a linear contraction on incremental scaling exponents -- is shown to be necessary and sufficient for the cascade multiplier to be log-Poisson. We establish three results: (1) a characterization theorem proving that the hierarchical symmetry uniquely determines the log-Poisson distribution with explicit parameters; (2) a classification theorem proving that the hierarchical symmetry selects exactly the log-Poisson class from the full log-infinitely-divisible family, excluding log-normal, log-stable, and all intermediate generators; and (3) a stability theorem proving that approximate hierarchical symmetry implies approximate log-Poisson, with an explicit Wasserstein bound. The proofs reduce the problem to the Hausdorff moment problem on via the change of variables , where determinacy and stability follow from classical results.

Paper Structure

This paper contains 6 sections, 12 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. The axiom
  4. Results
  5. Corollaries
  6. Concluding remarks

Key Result

Lemma 1

If the incremental exponents $\{\delta_p\}$ satisfy the A1 recurrence eq:A1 with $\beta \in (0,1)$, then with $\zeta_0 = 0$: 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. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Lemma 1: Exponent Form
  • proof
  • Lemma 2: Moment Determinacy
  • proof
  • Theorem 3: Characterization
  • proof
  • Remark : Conservation
  • Proposition 4: Converse
  • proof
  • Corollary 5: Biconditional
  • ...and 11 more