EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS

Götze F, Zaitsev AY. EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS. The Annals of Probability. 2014;42(1):354-397.Let X, X-1, X-2,... be i.i.d. R-d-valued real random vectors. Assume that EX = 0, cov X = C, E vertical bar vertical bar X vertical bar vertical bar(2) = sigma(2) and that X is not concentrated in a proper subspace of 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 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) =(def) sup(x)[P{Q[S-N] = 5 and the fourth moment of X exists. Furthermore, we provide explicit bounds of order O(N-1) for AN for the rate of approximation by short asymptotic expansions and for the concentration functions of the random variables Q[S-N + a], a epsilon R-d. The order of the bound is optimal. It extends previous results of Bentkus and Glaze [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 c(d) sigma(d) (detC)-E-1/2 vertical bar vertical bar C-1/2 X vertical bar vertical bar(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 Esseen [Acta Math. 77 (1945) 1-125]