Wigner chaos and the fourth moment

TL;DR

This paper establishes a fourth moment criterion for convergence to the semicircular law in free Wigner chaos, extending classical Gaussian results to free probability and providing new transfer principles and bounds.

Contribution

It introduces a fourth moment characterization for free Wigner integrals, extending Gaussian chaos results to free probability, and develops tools for comparing different chaos orders.

Findings

Convergence in law to semicircular distribution characterized by fourth moments.

New transfer principle linking free Wigner chaos to Gaussian Wiener chaos.

Quantitative bounds on distances between chaos orders.

Abstract

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer-Major theorem.