Fluctuations of Quadratic Chaos

TL;DR

This paper characterizes the distributional limits of quadratic forms of i.i.d. variables with Bernoulli coefficients, revealing they can be decomposed into Gaussian, chi-square, and Gaussian mixture components, and establishes a fourth moment criterion for normal convergence.

Contribution

It provides a comprehensive description of all possible limits of quadratic chaos with Bernoulli coefficients and proves a new fourth moment theorem applicable even without finite fourth moments.

Findings

Distributional limits are sums of Gaussian, chi-square, and Gaussian mixture components.

A fourth moment theorem characterizes asymptotic normality without requiring finite fourth moments.

Convergence to normality is equivalent to the fourth moment converging to 3.

Abstract

In this paper we characterize all distributional limits of the random quadratic form $T_n =\sum_{1\le u< v\le n} a_{u, v} X_u X_v$, where $((a_{u, v}))_{1\le u,v\le n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal and $X_1, X_2, \ldots, X_n$ are i.i.d.~ mean $0$ variance $1$ random variables with common distribution function $F$. In particular, we show that any distributional limit of $S_n:=T_n/\sqrt{\mathrm{Var}[T_n]}$ can be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite weighted sum of independent centered chi-squares, and a Gaussian mixture with a random variance. As a consequence, we prove a fourth moment theorem for the asymptotic normality of $S_n$, which applies even when $F$ does not have finite fourth moment. More formally, we show that $S_n$ converges to $N(0, 1)$ if and only if the fourth moment of $S_n$ (appropriately truncated when $F$ does not have finite fourth moment) converges to 3 (the fourth moment of the standard normal distribution).