A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core

2026/07/30 by Guangqu Zheng
Mathematics · #math.PR #msc:60F05 #msc:60H07 #msc:60G55 #msc:60H05

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arxiv created 2026/07/30 · arxiv updated 2026/08/03

Abstract

For any finite family of Poisson multiple integrals and any finite p≥2, we construct a common increasing filtration generated by finitely many exact Poisson counts such that the associated conditional expectations converge in L^p, remain in their original chaoses, and have bounded step kernels with finite-measure support. This finite-count martingale core allows regular fixed-chaos identities and estimates to be extended under the sole assumption of a finite fourth moment. In particular, if F lives in a Poisson chaos with unit variance and finite fourth moment, we prove that the Kolmogorov distance between F and a standard normal is bounded by 15.6(𝔼[F⁴]-3)^1/2. This removes Assumptions \mathbf A and \mathbf A^loc from the Kolmogorov bound of Döbler and Peccati (Ann. Probab., 2018). We also obtain quantitative L⁴ estimates for all iterated Malliavin derivatives and, for F in a Poisson chaos, the fourth moment assumption of F forces the L⁴-integrability of its kernel.

Citations

• The Poisson Multiplication Formula
• Fourth moment theorems on the Poisson space: analytic statements via\n product formulae
• The fourth moment theorem on the Poisson space
• Stochastic analysis for Poisson processes
• Normal approximation of Poisson functionals in Kolmogorov distance
• Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs
• Universal Gaussian fluctuations on the discrete Poisson chaos
• Multi-dimensional Gaussian fluctuations on the Poisson space
• Poisson process Fock space representation, chaos expansion and covariance inequalities
• Stein's method and normal approximation of Poisson functionals
• Stein's method on Wiener chaos
• Central limit theorems for sequences of multiple stochastic integrals