Classical and Free Fourth Moment Theorems: Universality and Thresholds

Abstract

Let \(X\) be a centered random variable with unit variance and zero third moment, and such that \(\mathrm{IE}[X^4] \ge 3\). Let \(\{F_n {:}\, n\ge 1\}\) denote a normalized sequence of homogeneous sums of fixed degree \(d\ge 2\), built from independent copies of \(X\). Under these minimal conditions, we prove that \(F_n\) converges in distribution to a standard Gaussian random variable if and only if the corresponding sequence of fourth moments converges to \(3\). The statement is then extended (mutatis mutandis) to the free probability setting. We shall also discuss the optimality of our conditions in terms of explicit thresholds, as well as establish several connections with the so-called universality phenomenon of probability theory. Both in the classical and free probability frameworks, our results extend and unify previous Fourth Moment Theorems for Gaussian and semicircular approximations. Our techniques are based on a fine combinatorial analysis of higher moments for homogeneous sums.