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

Ivan Nourdin (), Giovanni Peccati () and Xiaochuan Yang ()
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Ivan Nourdin: University of Luxembourg
Giovanni Peccati: University of Luxembourg
Xiaochuan Yang: University of Bath

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 2020-2037

Abstract: 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.

Keywords: Breuer–Major Theorem; Convex distance; Fourth moment theorems; Gaussian fields; Malliavin–Stein method; Multidimensional normal approximations; 60F05; 60G15; 60H07 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01112-6

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