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

TL;DR

This paper establishes an asymptotic version of Cramér's theorem on Wiener space, showing that the convergence of sums to a normal distribution implies individual convergence under independence, and compares it with Malliavin calculus criteria.

Contribution

It proves an asymptotic Cramér's theorem on Wiener space and relates it to recent Malliavin derivative-based convergence criteria.

Findings

Proves asymptotic Cramér's theorem for Wiener space.

Shows convergence of sums implies convergence of individual sequences.

Compares classical and Malliavin-based criteria for normal convergence.

Abstract

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 $n\geq 1$ the random variables $X_{n}$ and $Y_{n}$ are independent, then $(X_{n})_{n\geq 1}$ {\it and } $(Y_{n}) _{n\geq 1}$ 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.