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, cov X=C, E X²=σ 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, \[ΔN{def}{=}ₓ|P\{Q[SN]≤ x\}-P\{Q[G]≤ x\}|={O}(N⁻¹),\] provided that d≥5 and the fourth moment of X exists. Furthermore, we provide explicit bounds of order O(N⁻¹) for ΔN 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 cdσ ᵈ()-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].