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.
Categories: math.PR