Convergence in total variation on Wiener chaos
Résumé
Let $\{F_n\}$ be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards $F_\infty$ satisfying ${\rm Var}(F_\infty)>0$. Our first result is a sequential version of a theorem by Shigekawa \cite{Shigekawa}. More precisely, we prove, without additional assumptions, that the sequence $\{F_n\}$ actually converges in total variation and that the law of $F_\infty$ is absolutely continuous. We give an application to discrete non-Gaussian chaoses. In a second part, we assume that each $F_n$ has more specifically the form of a multiple Wiener-Itô integral (of a fixed order) and that it converges in $L^2(\Omega)$ towards $F_\infty$. We then give an upper bound for the distance in total variation between the laws of $F_n$ and $F_\infty$. As such, we recover an inequality due to Davydov and Martynova \cite{DM}; our rate is weaker compared to \cite{DM} (by a power of 1/2), but the advantage is that our proof is not only sketched as in \cite{DM}. Finally, in a third part we show that the convergence in the celebrated Peccati-Tudor theorem actually holds in the total variation topology.
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