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Convergence in total variation on Wiener chaos

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  • Nourdin, Ivan
  • Poly, Guillaume

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.

Suggested Citation

  • Nourdin, Ivan & Poly, Guillaume, 2013. "Convergence in total variation on Wiener chaos," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 651-674.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:2:p:651-674
    DOI: 10.1016/j.spa.2012.10.004
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    References listed on IDEAS

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    1. Breton, Jean-Christophe, 2006. "Convergence in variation of the joint laws of multiple Wiener-Itô integrals," Statistics & Probability Letters, Elsevier, vol. 76(17), pages 1904-1913, November.
    2. Nualart, D. & Ortiz-Latorre, S., 2008. "Central limit theorems for multiple stochastic integrals and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 614-628, April.
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    Cited by:

    1. Nourdin, Ivan & Diu Tran, T.T., 2019. "Statistical inference for Vasicek-type model driven by Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3774-3791.
    2. Es-Sebaiy, Khalifa & Viens, Frederi G., 2019. "Optimal rates for parameter estimation of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3018-3054.
    3. Nourdin, Ivan & Poly, Guillaume, 2015. "An invariance principle under the total variation distance," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2190-2205.
    4. Loosveldt, L., 2023. "Multifractional Hermite processes: Definition and first properties," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 465-500.
    5. Nourdin, Ivan & Nualart, David & Peccati, Giovanni, 2021. "The Breuer–Major theorem in total variation: Improved rates under minimal regularity," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 1-20.
    6. Luca Pratelli & Pietro Rigo, 2018. "Convergence in Total Variation to a Mixture of Gaussian Laws," Mathematics, MDPI, vol. 6(6), pages 1-14, June.
    7. Davydov, Youri, 2017. "On distance in total variation between image measures," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 393-400.

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