On almost sure convergence of random variables with finite chaos decomposition

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 Wiener-Itô integrals with respect to Poisson processes, answering a question by Poly and Zheng.