Convergence in total variation on Wiener chaos

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Let Fₙ be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards F∞ satisfying Var(F∞)>0. Our first result is a sequential version of a theorem by Shigekawa (1980). More precisely, we prove, without additional assumptions, that the sequence Fₙ actually converges in total variation and that the law of F∞ is absolutely continuous. We give an application to discrete non-Gaussian chaoses. In a second part, we assume that each Fₙ has more specifically the form of a multiple Wiener-Itô integral (of a fixed order) and that it converges in L²(Ω) towards F∞. We then give an upper bound for the distance in total variation between the laws of Fₙ and F∞. 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.