Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1155619 | Stochastic Processes and their Applications | 2013 | 24 Pages |
Abstract
Let {Fn} be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards Fâ satisfying V ar(Fâ)>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 {Fn} actually converges in total variation and that the law of Fâ 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 Fâ. We then give an upper bound for the distance in total variation between the laws of Fn and Fâ. 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Ivan Nourdin, Guillaume Poly,