Classical and free Fourth Moment Theorems: universality and thresholds

TL;DR

This paper establishes a minimal set of conditions under which homogeneous sums of independent variables converge to a Gaussian or semicircular distribution, extending classical and free probability Fourth Moment Theorems.

Contribution

It proves a new Fourth Moment Theorem under minimal moment conditions and extends it to free probability, unifying classical and free cases with optimal thresholds.

Findings

Convergence to Gaussian occurs if and only if fourth moments converge to 3.

Results extend previous Fourth Moment Theorems for Gaussian and semicircular laws.

Provides thresholds and conditions for universality in classical and free probability.

Abstract

Let $X$ be a centered random variable with unit variance, zero third moment, and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized sequence of homogeneous sums of fixed degree $d\geq 2$, 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.