Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces

Summary.   Let X,X ₁,X ₂,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ ^d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ ^d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S _N ] of the normalized sums S _N =N ^−1/2(X ₁+⋯+X _N ) and show that provided that d≥9 and the fourth moment of X exists. The bound ?(N ⁻¹) is optimal and improves, e.g., the well-known bound ?(N ^{− d /( d +1)}) due to Esseen (1945). The result extends to the case of random vectors taking values in a Hilbert space. Furthermore, we provide explicit bounds for Δ _N and for the concentration function of the random variable ℚ[S _N ]. Received: 9 January 1997 / In revised form: 15 May 1997