Uniform rates of approximation by short asymptotic expansions 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 $\mathbf EX=0$ and that $X$ has a non-degenerate distribution. Let $G$ be a mean zero Gaussian random vector with the same covariance operator as that of $X$. We investigate the distributions of non-degenerate quadratic forms $\mathbb Q[S_N]$ of the normalized sums $S_N=N^{-1/2}(X_1+\dots+X_N)$ and show that, without any additional conditions, for any $a\in\mathbb R^d$,
$$\Delta_N^{(a)}\stackrel{\mathrm{def}}=\sup_x\bigl|\mathbf P\bigl\{\mathbb Q[S_N-a]\le x\bigr\}-\mathbf P\bigl\{\mathbb Q[G-a]\le x\bigr\}-E_a(x)\bigr|=\mathcal O\bigl(N^{-1}\bigr),$$
provided that $d\ge5$ and $\mathbf E\left\|X\right\|^4<\infty$. Here $E_a(x)$ is the Edgeworth type correction of order $\mathcal O\bigl(N^{-1/2}\bigr)$. Furthermore, we provide explicit bounds of order $\mathcal O\bigl(N^{-1}\bigr)$ for $\Delta_N^{(a)}$ and for the concentration function of the random variable $\mathbb Q[S_N+a]$, $a\in\mathbb R^d$. Our results extend the corresponding results of Bentkus and Götze (1997) ($d\ge9$) to the case $d\ge5$. Bibl. 35 titles.