Gaussian Approximations of Multiple Integrals

TL;DR

This paper provides minimal-assumption criteria for when sequences of multiple Wiener-Itô integrals converge in distribution to Gaussian vectors, generalizing previous results and aiding the analysis of high-frequency stationary fields.

Contribution

It establishes necessary and sufficient conditions for Gaussian approximations of multiple Wiener-Itô integrals under minimal assumptions, extending prior work.

Findings

Conditions for asymptotic Gaussianity without covariance convergence

Minimal assumptions on variances and covariances

Generalization of earlier Gaussian approximation results

Abstract

Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-dimensional vectors of multiple Wiener-It\^o integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l diverges, the law of I(l) is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as I(l). The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of I(l). In particular, we will not assume that the covariance matrix of I(l) is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.