Convergence in total variation on Wiener chaos

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:123

Article

Nourdin, Ivan ; Poly, Guillaume

关键词: Convergence in distribution;    Convergence in total variation;    Malliavin calculus;    Multiple Wiener-Ito integral;    Wiener chaos;   

DOI  :  10.1016/j.spa.2012.10.004

来源: Elsevier

【 摘 要 】

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-infinity satisfying Var(F-infinity) > 0. Our first result is a sequential version of a theorem by Shigekawa (1980) [23]. More precisely, we prove, without additional assumptions, that the sequence {F-n} actually converges in total variation and that the law of F-infinity 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-Ito integral (of a fixed order) and that it converges in L-2(Omega) towards F-infinity. We then give an upper bound for the distance in total variation between the laws of F-n and F-infinity. As such, we recover an inequality due to Davydov and Martynova (1987) [5]; our rate is weaker compared to Davydov and Martynova (1987) [5] (by a power of 1/2), but the advantage is that our proof is not only sketched as in Davydov and Martynova (1987) [5]. Finally, in a third part we show that the convergence in the celebrated Peccati-Tudor theorem actually holds in the total variation topology. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free