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 ℝd. Assume that E X=0 and that X is not concentrated in a proper subspace of ℝd. 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 ].
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Received: 9 January 1997 / In revised form: 15 May 1997
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Bentkus, V., Götze, F. Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces. Probab Theory Relat Fields 109, 367–416 (1997). https://doi.org/10.1007/s004400050136
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DOI: https://doi.org/10.1007/s004400050136