Multivariate Gaussian approximations on Markov chaoses

TL;DR

This paper extends the multidimensional Fourth Moment Theorem to chaotic random vectors within diffusion Markov generators, requiring an additional moment condition for joint convergence beyond componentwise convergence.

Contribution

It introduces a generalized version of the Fourth Moment Theorem applicable to diffusion Markov generators, highlighting the need for an extra moment condition for joint convergence.

Findings

Proves a multidimensional Fourth Moment Theorem for chaotic vectors

Identifies an additional moment condition for joint convergence

Extends results beyond the Ornstein-Uhlenbeck case

Abstract

We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.