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Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

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  • Ivan Nourdin

    (University of Luxembourg)

  • Giovanni Peccati

    (University of Luxembourg)

  • Xiaochuan Yang

    (University of Bath)

Abstract

We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

Suggested Citation

  • Ivan Nourdin & Giovanni Peccati & Xiaochuan Yang, 2022. "Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2020-2037, September.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01112-6
    DOI: 10.1007/s10959-021-01112-6
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    References listed on IDEAS

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    1. de Jong, Peter, 1990. "A central limit theorem for generalized multilinear forms," Journal of Multivariate Analysis, Elsevier, vol. 34(2), pages 275-289, August.
    2. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    3. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107039124, September.
    4. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986, September.
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