Chaos of a Markov operator and the fourth moment condition

TL;DR

This paper explores the relationship between Markov operators and the fourth moment condition in Wiener chaos, providing new spectral conditions for convergence to Gaussian and gamma distributions.

Contribution

It introduces a notion of chaos linked to Markov operators and establishes spectral conditions for chaos eigenfunctions to converge to Gaussian or gamma distributions.

Findings

Spectral conditions for Gaussian convergence of chaos eigenfunctions

Extension of fourth moment criteria to Markov operator framework

Analysis of convergence to gamma distributions

Abstract

We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly.