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arXiv · 2001.02188

Multivariate normal approximation on the Wiener space: new bounds in the convex distance

Ivan Nourdin·Giovanni Peccati·Xiaochuan Yang
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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, Peccati and R\'eveillac (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 (2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener-It\^o integrals, and (ii) we characterise the rate of convergence for the finite-dimensional distributions in the functional Breuer-Major theorem.

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Ivan Nourdin, Giovanni Peccati, Xiaochuan Yang. 2020-01-07. Multivariate normal approximation on the Wiener space: new bounds in the convex distance. https://arxiv.org/abs/2001.02188

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