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Convergence in total variation on Wiener chaos

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  • Nourdin, Ivan
  • Poly, Guillaume
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    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 F∞ satisfying V ar(F∞)>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 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) [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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 123 (2013)
    Issue (Month): 2 ()
    Pages: 651-674

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    Handle: RePEc:eee:spapps:v:123:y:2013:i:2:p:651-674

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    Related research

    Keywords: Convergence in distribution; Convergence in total variation; Malliavin calculus; Multiple Wiener–Itô integral; Wiener chaos;

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