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Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

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  • Eden, Richard
  • Víquez, Juan

Abstract

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.

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  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:182-216
    DOI: 10.1016/j.spa.2014.09.001
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    References listed on IDEAS

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    1. Viens, Frederi G., 2009. "Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3671-3698, October.
    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.
    3. 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.
    4. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.
    5. Kusuoka, Seiichiro & Tudor, Ciprian A., 2012. "Stein’s method for invariant measures of diffusions via Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1627-1651.
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    Cited by:

    1. Gaunt, Robert E., 2019. "Stein operators for variables form the third and fourth Wiener chaoses," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 118-126.
    2. Arras, Benjamin & Azmoodeh, Ehsan & Poly, Guillaume & Swan, Yvik, 2019. "A bound on the Wasserstein-2 distance between linear combinations of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2341-2375.
    3. Robert E. Gaunt, 2020. "Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I," Journal of Theoretical Probability, Springer, vol. 33(1), pages 465-505, March.

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