Explicit rates of approximation in the CLT for quadratic forms

TL;DR

This paper establishes explicit, optimal rates of convergence in the Central Limit Theorem for quadratic forms of sums of i.i.d. random vectors in high dimensions, extending previous results to the case d≥5.

Contribution

It provides the first explicit bounds of order N^{-1} for the approximation of quadratic forms in the CLT for dimensions as low as 5, with precise constants.

Findings

Rate of convergence is O(N^{-1}) for quadratic forms in high dimensions.

Explicit bounds depend on the fourth moment and covariance structure.

Extends previous results to lower dimensions (d≥5).

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 G\"{o}tze [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 Ess\'{e}en [Acta Math. 77 (1945) 1-125].