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On the Gaussian approximation of vector-valued multiple integrals

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  • Noreddine, Salim
  • Nourdin, Ivan

Abstract

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals Fn towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of Fn to zero; (ii) the covariance matrix of Fn to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a -valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.

Suggested Citation

  • Noreddine, Salim & Nourdin, Ivan, 2011. "On the Gaussian approximation of vector-valued multiple integrals," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1008-1017, July.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:6:p:1008-1017
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    References listed on IDEAS

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    1. Fox, Robert & Taqqu, Murad S., 1987. "Multiple stochastic integrals with dependent integrators," Journal of Multivariate Analysis, Elsevier, vol. 21(1), pages 105-127, February.
    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. Yoon-Tae Kim & Hyun-Suk Park, 2022. "Fourth Cumulant Bound of Multivariate Normal Approximation on General Functionals of Gaussian Fields," Mathematics, MDPI, vol. 10(8), pages 1-17, April.
    2. Kim, Yoon Tae & Park, Hyun Suk, 2018. "An Edgeworth expansion for functionals of Gaussian fields and its applications," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 3967-3999.
    3. Eden, Richard & Víquez, Juan, 2015. "Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 182-216.

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