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Preprint, Working Paper, ... |
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| Subject: |
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Mathematics/Probability
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| Title: |
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Convergence in total variation on Wiener chaos |
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| Author(s): |
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Ivan Nourdin ( ,  ) 1, Guillaume Poly () 2 |
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| Laboratory: |
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| Abstract: |
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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 P(F_infinity=0)<1. Our first result is a sequential version of a theorem by Shigekawa. More precisely, we prove that the sequence {F_n} actually converges in total variation and that the law of F_infinity is absolutely continuous. In a second part, we assume that each F_n has more specifically the form of a multiple Wiener-Itô integral (of a fixed order) and that it converges in L^2 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; our rate is a bit weaker compared to them, but the advantage is that our proof is not only sketched. 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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| Fulltext language: |
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English |
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| Keyword(s): |
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Convergence in law – Convergence in total variation – Malliavin calculus – multiple Wiener-Itô integral – Wiener chaos. |
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| Comment: |
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23 pages |
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