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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.01632 |
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| _version_ | 1866914439415988224 |
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| author | Freeburg, E. M. |
| author_facet | Freeburg, E. M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01632 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability Freeburg, E. M. Probability Mathematical Physics 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. |
| title | Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2604.01632 |