Yet another proof of the Nualart-Peccati criterion

TL;DR

This paper provides a simple, unified proof of the Nualart-Peccati criterion and its extension to free probability, showing that convergence of the fourth moment characterizes Gaussian convergence for multiple integrals.

Contribution

It offers an elementary, unified proof of the Nualart-Peccati criterion and its extension to free Brownian motion, using only basic combinatorial arguments and the product formula.

Findings

Unified proof of the classical and free probability versions of the criterion.

The proof relies solely on the product formula and combinatorial arguments.

The criterion characterizes Gaussian convergence via the fourth moment.

Abstract

In 2005, Nualart and Peccati showed that, surprisingly, the convergence in distribution of a normalized sequence of multiple Wiener-It\^o integrals towards a standard Gaussian law is equivalent to convergence of just the fourth moment to 3. Recently, this result has been extended to a sequence of multiple Wigner integrals, in the context of free Brownian motion. The goal of the present paper is to offer an elementary, unifying proof of these two results. The only advanced, needed tool is the product formula for multiple integrals. Apart from this formula, the rest of the proof only relies on soft combinatorial arguments.