Fourth Moment Theorems for Markov Diffusion Generators

TL;DR

This paper revisits the Fourth Moment Theorem for Markov diffusion generators, simplifying proofs and extending the criterion to new settings like Laguerre and Jacobi, with discussions on convergence to gamma and beta distributions.

Contribution

It simplifies existing proofs of the Fourth Moment Theorem and extends its applicability to Laguerre and Jacobi diffusive generators.

Findings

Simplified proof of the Fourth Moment Theorem.

Extended the theorem to Laguerre and Jacobi generators.

Discussed convergence to gamma and beta distributions.

Abstract

Inspired by the insightful article arXiv:1210.7587, we revisit the Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.