Asymptotic Cramér's theorem and analysis on Wiener space

Abstract

We prove an asymptotic Cramér's theorem, that is, if the sequence $(X_n+Y_n{)}_{n\ge 1}$ converges in law to the standard normal distribution and for every $n\ge 1$ the random variables $X_n$ and $Y_n$ are independent, then $(X_n{)}_{n\ge 1}$ {\it and } $(Y_n{)}_{n\ge 1}$ converge in law to a normal distribution. Then we compare this result with recent criteria for the central convergence obtained in terms of Malliavin derivatives.

To appear in "Seminaire de Probabilites XLIII"