Four moments theorems on Markov chaos

TL;DR

This paper establishes quantitative Four Moments Theorems for Markov chaos, showing convergence to Pearson distributions based on four moments, with a novel bound on the law distance using Markov diffusion generators.

Contribution

It introduces a new approach to prove convergence to Pearson distributions using only four moments for elements of Markov chaos, extending moment-based convergence results.

Findings

Convergence of Markov chaos elements to Pearson laws with four moments.

A general carré du champ bound on the law distance in Markov diffusions.

Reduction of the bound to the first four moments for elements of Markov chaos.

Abstract

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carr\'e du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.