Almost sure convergence on chaoses

TL;DR

This paper investigates almost sure convergence in homogeneous chaoses, showing that convergence of a finite sum implies convergence of each component, using conditioning and exchangeable pairs techniques.

Contribution

It introduces new phenomena and methods for analyzing almost sure convergence in homogeneous chaoses, including Gaussian and independent sums.

Findings

Almost sure convergence on a finite sum implies convergence of each chaos component.

Uses conditioning and exchangeable pairs techniques.

Leaves open questions for further research.

Abstract

We present several new phenomena about almost sure convergence on homogeneous chaoses that include Gaussian Wiener chaos and homogeneous sums in independent random variables. Concretely, we establish the fact that almost sure convergence on a fixed finite sum of chaoses forces the almost sure convergence of each chaotic component. Our strategy uses "{\it extra randomness}" and a simple conditioning argument. These ideas are close to the spirit of \emph{Stein's method of exchangeable pairs}. Some natural questions are left open in this note.