On the structure of Gaussian random variables

TL;DR

This paper investigates conditions under which Gaussian random variables are almost surely equal to Brownian motion variables and explores implications for the distribution of variables in Wiener chaoses, advancing understanding in Malliavin calculus.

Contribution

It provides new criteria for Gaussian variables to coincide with Brownian motion and shows that certain Wiener chaos variables cannot be normally distributed.

Findings

Gaussian variables equal to Brownian motion are characterized.

Variables in finite Wiener chaoses cannot be Gaussian under certain conditions.

Enhanced understanding of Gaussian characterization via Malliavin calculus.

Abstract

We study when a given Gaussian random variable on a given probability space $(\Omega, {\cal{F}}, P) $ is equal almost surely to $\beta_{1}$ where $\beta $ is a Brownian motion defined on the same (or possibly extended) probability space. As a consequences of this result, we prove that the distribution of a random variable (satisfying in addition a certain property) in a finite sum of Wiener chaoses cannot be normal. This result also allows to understand better some characterization of the Gaussian variables obtained via Malliavin calculus.