A Peccati-Tudor type theorem for Rademacher chaoses

TL;DR

This paper establishes a new quantitative normal approximation result for Rademacher chaoses, showing that small influences and cumulants imply convergence to Gaussian vectors, extending previous univariate results.

Contribution

It introduces a novel adaptation of exchangeable pairs coupling combined with chaos decomposition estimates for multivariate Rademacher chaoses.

Findings

Close approximation to Gaussian vectors when influences and cumulants are small

Extension of univariate results to multivariate Rademacher chaoses

New coupling method for chaos-based normal approximation

Abstract

In this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centred Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in D\"obler and Krokowski (2017). Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition.