Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces |
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Authors: | V Bentkus F Götze |
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Institution: | Fakult?t für Mathematik, Universit?t Bielefeld, Postfach 100131, D-33501 Bielefeld 1, Germany E-mail addresses: bentkus@mathematik.uni-bielefeld.de; goetze@mathematik.uni-bielefeld.de, DE
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Abstract: | 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 ℚS
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 ℚS
N
].
Received: 9 January 1997 / In revised form: 15 May 1997 |
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Keywords: | AMS Subject Classification (1991): Primary 60F05 secondary 62E20 |
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