Explicit rates of approximation in the CLT for quadratic forms

Let X,X1,X2,. be i.i.d. Rd-valued real random vectors. Assume that EX=0, covX=C, E.X.2=.2 and that X is not concentrated in a proper subspace of Rd. 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 Q[SN] of the normalized sums SN=N.1/2(X1+.+XN) and show that, without any additional conditions, .Ndef=supx..P{Q[SN].x}.P{Q[G].x}..=O(N.1), provided that d.5 and the fourth moment of X exists. Furthermore, we provide explicit bounds of order O(N.1) for .N for the rate of approximation by short asymptotic expansions and for the concentration functions of the random variables Q[SN+a], a.Rd. The order of the bound is optimal. It extends previous results of Bentkus and Götze [Probab. Theory Related Fields 109 (1997a) 367.416] (for d.9) to the case d.5, which is the smallest possible dimension for such a bound. Moreover, we show that, in the finite dimensional case and for isometric Q, the implied constant in O(N.1) has the form cd.d(detC).1/2E.C.1/2X.4 with some cd 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 Esséen [Acta Math. 77 (1945) 1.125].