On almost sure convergence of random variables with finite chaos decomposition

TL;DR

This paper proves that for independent random variables, almost sure convergence of polynomial chaos sequences implies convergence of their components, generalizing previous results and providing conditions for i.i.d. sequences and Poisson integrals.

Contribution

It extends almost sure convergence results for polynomial chaos to broader settings, including i.i.d. sequences and Poisson processes, with weaker assumptions than prior work.

Findings

Almost sure convergence of chaos implies convergence of homogeneous parts.

Provides necessary and sufficient conditions for i.i.d. sequences.

Discusses convergence phenomena for Poisson stochastic integrals.

Abstract

Under mild conditions on a family of independent random variables $(X_n)$ we prove that almost sure convergence of a sequence of tetrahedral polynomial chaoses of uniformly bounded degrees in the variables $(X_n)$ implies the almost sure convergence of their homogeneous parts. This generalizes a recent result due to Poly and Zheng obtained under stronger integrability conditions. In particular for i.i.d. sequences we provide a simple necessary and sufficient condition for this property to hold. We also discuss similar phenomena for sums of multiple stochastic integrals with respect to Poisson processes, answering a question by Poly and Zheng.