Explicit rates of approximation in the CLT for quadratic forms

Let $X,X_1,X_2,\dots$ be i.i.d. ${\mathbb{R}}^d$-valued real random vectors. Assume that $\mathbf{E}X=0$, $\mathrm{cov}X=\mathbb{C}$, $\mathbf{E}\mathrm{\parallel }X{\mathrm{\parallel }}^2={\sigma }^2$ 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 $S_N=N^{-1\mathrm{/}2}(X_1+\cdots +X_N)$ and show that, without any additional conditions, $${\mathrm{\Delta }}_N\underset{x}{\sup }\mid \mathbf{P}\{\mathbb{Q}[S_N]\le x\}-\mathbf{P}\{\mathbb{Q}[G]\le x\}\mid =\mathcal{O}\left(N^{-1}\right),$$ provided that $d\ge 5$ and the fourth moment of $X$ exists. Furthermore, we provide explicit bounds of order $\mathcal{O}(N^{-1})$ for ${\mathrm{\Delta }}_N$ 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 $d\ge 9$) 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}$)^{-1/2}\mathbf {E}|$\mathbb{C}$^{-1/2}X|^4$withsome$c_d$dependingon$d$ 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].