Explicit rates of approximation in the CLT for quadratic forms

Details Explicit rates of approximation in the CLT for quadratic forms Explicit rates of approximation in the CLT for quadratic forms

2011-04-04

arxiv.org/abs/1104.0519v1

Mathematics

Probability

Scientific paper

Let $X, X_1, X_2, ... $ be {i.i.d.} ${\mathbb R}^d$-valued real random vectors. Assume that ${\bf E}\, X=0\,$, $\cov X =\mathbb C$, ${\bf E}\, \norm X^2=\s^2\,$ 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 non-degenerate quadratic forms $ \mathbb Q [S_N ] $ of the normalized sums ${S_N=N^{-1/2}\, (X_1+...+X_N)}\, $ and show that, without any additional conditions, $$\Delta_N\= \sup_x\, \Bigl|{\bf P}\bigl\{\mathbb Q \4[S_N]\leq x\bigr\}- {\bf P}\bigl\{\mathbb Q \4[G]\leq x\bgr\}\Bgr| = {\mathcal O}\bigl(N^{-1}\bigr),$$ provided that $d\geq 5\, $ and the fourth moment of $X\, $ exists. Furthermore, we provide explicit bounds of order ${\mathcal O}\bgl(N^{-1}\bgr)$ 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\"otze (1997a) (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 ${\mathcal O}\bgl(N^{-1}\bgr)$ has the form $c_d\, \s^d\,(\det \mathbb C)^{-1/2}\,\E\|\mathbb C^{-1/2}\,X\|^4$ 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 C.-G. Esseen (1945).

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