Explicit rates of approximation in the CLT for quadratic forms

Abstract

Let $X,X_{1},X_{2},\ldots$ be i.i.d. ${\mathbb{R}}^{d}$-valued real random vectors. Assume that ${\mathbf{E}X=0}$, $\operatorname{cov} X=\mathbb{C}$, $\mathbf{E}\Vert X\Vert^{2}=\sigma ^{2}$ 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,

\[\Delta_{N}\stackrel{\mathrm{def}}{=}\sup_{x}\bigl|\mathbf{P}\bigl\{\mathbb{Q}[S_{N}]\leq x\bigr\}-\mathbf{P}\bigl\{\mathbb{Q}[G]\leq x\bigr\}\bigr|={\mathcal{O}}\bigl(N^{-1}\bigr),\]

provided that $d\geq5$ 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 ${d\ge9}$) to the case $d\ge5$, 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 $c_{d}\sigma ^{d}(\det\mathbb{C})^{-1/2}\mathbf{E} \|\mathbb{C}^{-1/2}X\|^{4}$ 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].