# On the structure of Gaussian random variables

1 CES - Centre d-économie de la Sorbonne 2 SAMOS - Statistique Appliquée et MOdélisation Stochastique

Abstract : We study when a given Gaussian random variable on a given probability space $\left \Omega , {\cal{F}}, P ight$ 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.

Author: Ciprian Tudor -

Source: https://hal.archives-ouvertes.fr/