Abstract
AbstractWe 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.
Highlights
Fix m ≥ 1 and consider random vectors F and G with values in Rm
The aim of the present note is to establish explicit bounds on the quantity dc(F, G), in the special case where F is a vector of smooth functionals of an infinite-dimensional Gaussian field, and G = N is a m-dimensional centered Gaussian vector with covariance > 0
Our main tool is the so-called Malliavin–Stein method for probabilistic approximations [17], that we will combine with some powerful recursive estimates on dc, recently derived in [29] in the context of multidimensional second-order Poincaré inequalities on the Poisson space—see Lemma 2.1
Summary
Fix m ≥ 1 and consider random vectors F and G with values in Rm. The convex distance between the distributions of F and G is defined as dc(F, G) := sup Eh(F) − Eh(G) ,. The specific challenge we are setting ourselves in the present work is to establish bounds on the quantity dc(F, N ) that coincide (up to an absolute multiplicative constant) with the bounds deduced in [19] on the 1-Wasserstein distance dW (F, N ) := sup |Eh(F) − Eh(N )| ,. Remark 1.1 In order for the quantity dW (F, N ) to be well-defined, one needs that E F Rm < ∞. For p ≥ 1, we write L p( ) := L p( , F , P)
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