Fluctuations of Quadratic Chaos

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 $X_1,X_2,\dots ,X_n$ 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).

43 pages, 3 figures