Explicit rates of approximation in the CLT for quadratic forms

Abstract

Let $X,X_1,X_2,\dots$ be i.i.d. ${\mathbb{R}}^d$-valued real random vectors. Assume that $\mathbf{E}X=0$, $\mathrm{cov}X=\mathbb{C}$, $\mathbf{E}\Vert X{\Vert }^2={\sigma }^2$ and that $X$ is not concentrated in a proper subspace of ${\mathbb{R}}^d$. Let $G$ be a mean zero Gaussian random vector with the same covariance operator as that of $X$. We study the distributions of nondegenerate quadratic forms $\mathbb{Q}[S_N]$ of the normalized sums $S_N=N^{-1/2}(X_1+\cdots +X_N)$ and show that, without any additional conditions, \[Δ_N\stackrel{\mathrm{def}}{=}\sup_x\bigl |\mathbf{P}\bigl\{\mathbb{Q}[S_N]\leq x\bigr\}-\mathbf{P}\bigl\{\mathbb{Q}[G]\leq x\bigr\}\bigr|={\mathcal{O}}\bigl(N^{-1}\bigr),\] provided that $d\ge 5$ and the fourth moment of $X$ exists. Furthermore, we provide explicit bounds of order $\mathcal{O}(N^{-1})$ for ${\Delta }_N$ for the rate of approximation by short asymptotic expansions and for the concentration functions of the random variables $\mathbb{Q}[S_N+a]$, $a\in {\mathbb{R}}^d$. The order of the bound is optimal. It extends previous results of Bentkus and Götze [Probab. Theory Related Fields 109 (1997a) 367-416] (for $d\ge 9$) to the case $d\ge 5$, which is the smallest possible dimension for such a bound. Moreover, we show that, in the finite dimensional case and for isometric $\mathbb{Q}$, the implied constant in $\mathcal{O}(N^{-1})$ has the form $c_d{\sigma }^d(\det \mathbb{C}{)}^{-1/2}\mathbf{E}\Vert {\mathbb{C}}^{-1/2}X{\Vert }^4$ with some $c_d$ depending on $d$ only. This answers a long standing question about optimal rates in the central limit theorem for quadratic forms starting with a seminal paper by Esséen [Acta Math. 77 (1945) 1-125].

Published in at http://dx.doi.org/10.1214/13-AOP839 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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