Asymptotic Cram\'er's theorem and analysis on Wiener space

We prove an asymptotic Cram'er’s theorem, that is, if the sequence $(X_{n}+ Y_{n})_{n\geq 1}$ 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.