Convergence in total variation on Wiener chaos

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 (1980). 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-It\^o 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 (1987); our rate is weaker compared to Davydov and Martynova (1987) (by a power of 1/2), but the advantage is that our proof is not only sketched as in Davydov and Martynova (1987). Finally, in a third part we show that the convergence in the celebrated Peccati-Tudor theorem actually holds in the total variation topology.