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

Bentkus V, Götze F (1997)
PROBABILITY THEORY AND RELATED FIELDS 109(3): 367-416.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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].
Stichworte
concentration inequalities; convergence rates; Berry-Esseen bounds; Edgeworth expansions; multidimensional spaces; Hilbert spaces; quadratic forms; point problem; lattice; hyperboloids; ellipsoids; Central Limit Theorem
Erscheinungsjahr
1997
Zeitschriftentitel
PROBABILITY THEORY AND RELATED FIELDS
Band
109
Ausgabe
3
Seite(n)
367-416
ISSN
0178-8051
Page URI
https://pub.uni-bielefeld.de/record/1626698

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Bentkus V, Götze F. Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces. PROBABILITY THEORY AND RELATED FIELDS. 1997;109(3):367-416.
Bentkus, V., & Götze, F. (1997). Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces. PROBABILITY THEORY AND RELATED FIELDS, 109(3), 367-416.
Bentkus, V., and Götze, F. (1997). Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces. PROBABILITY THEORY AND RELATED FIELDS 109, 367-416.
Bentkus, V., & Götze, F., 1997. Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces. PROBABILITY THEORY AND RELATED FIELDS, 109(3), p 367-416.
V. Bentkus and F. Götze, “Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces”, PROBABILITY THEORY AND RELATED FIELDS, vol. 109, 1997, pp. 367-416.
Bentkus, V., Götze, F.: Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces. PROBABILITY THEORY AND RELATED FIELDS. 109, 367-416 (1997).
Bentkus, V, and Götze, Friedrich. “Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces”. PROBABILITY THEORY AND RELATED FIELDS 109.3 (1997): 367-416.