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

Friedrich Götze, Andrei Yu. Zaitsev
2014 Annals of Probability
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,
more » ... )}$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].