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 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].Comment: 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

A multivariate CLT for <> weighted sums with rate of convergence of order O(1/n)

“…), where 2 = −1 ( see (8)), it follows that = . Using Holder's inequality, we obtain that for the characteristic function of the random vector at ≤ − 1 the following inequality holds…”

“…, where the matrix is defined in (8). Also, taking into account that for any ≥ 4 from (16) it follows…”

High-dimensional central limit theorems by Stein’s method

“…For Euclidean balls centered at 0, it is known that (cf. Götze and Zaitsev (2014)) the dependence on n may be improved from 1/ √ n to 1/n for d ≥ 5, which is in general the smallest possible dimension for such an improvement. See also Esseen (1945), Bentkus and Götze (1997), Götze and Ulyanov (2003), Bogatyrev, Götze and Ulyanov (2006) and Prokhorov and Ulyanov (2013) for earlier and related results.…”

Inference via Randomized Test Statistics

“…The notation O r emphasizes that the hidden constant depends on r. To our knowledge, there are no similar bounds for d K (Law (T J ) , χ 2 (r − 1)) in the literature. Fortunately, one can derive even better rates of convergence than O r (n −1+1/r ) using recent advances in CLT for quadratic forms [17,Theorem 1.2] in addition to the classical result [9, Chapter 7, Theorem 1]. We provide the details in Proposition 2.3 below.…”

“…The main obstacle in the analysis of T φ λ , λ = 1, is that, in contrast to Pearson's statistic, it is not a quadratic form anymore, and consequently, the results of Götze [15, Theorem 1.5] or of Götze and Zaitsev [17,Theorem 1.2] are not applicable anymore. Due to Taylor's expansion, the phi-divergence test statistic T φ can be represented in the form…”