On the almost sure convergence of sums

TL;DR

This paper presents counterexamples related to almost sure convergence in chaoses, showing that certain convergence properties do not imply almost sure convergence of individual components.

Contribution

It provides counterexamples demonstrating that convergence in probability and in L_{2+δ} does not guarantee almost sure convergence for components in chaoses.

Findings

Counterexamples show failure of almost sure convergence

Convergence in probability and L_{2+δ} does not imply a.s. convergence

Components in chaoses can fail to converge almost surely despite other convergences

Abstract

Two counterexamples, addressing questions raised in \cite{AD} and \cite{PZ}, are provided. Both counterexamples are related to chaoses. Let $F_n=Y_n+Z_n$. It may be that $F_n\overset{a.s.}\longrightarrow 0$, $F_n\overset{L_{2+\delta}}\longrightarrow 0$ and $E\bigl\{\sup_n\,\abs{F_n}^\delta\bigr\}<\infty$, where $\delta>0$ and $Y_n$ and $Z_n$ belong to chaoses of uniformly bounded degree, and yet $Y_n$ fails to converge to 0 a.s.