Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces
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 EX = 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 [GRAPHICS] provided that d greater than or equal to 9 and the fourth moment of X exists. The bound O(N-1) is optimal and improves, e.g., the well-known bound O(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 Delta(N) and for the concentration function of the random variable Q[S-N].
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367-416
367-416
SPRINGER VERLAG