### Optimal bounds in non-Gaussian limit theorems for U-statistics

Bentkus V, Götze F (1999)
ANNALS OF PROBABILITY 27(1): 454-521.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Autor*in
Einrichtung
Abstract / Bemerkung
Let X, X-1, X-2,... be i.i.d. random variables taking values in a measurable space H. Let phi(x, y) and phi(1)(x) denote measurable functions of the arguments x, y is an element of H. Assuming that the kernel phi is symmetric and that E phi(x, X) = 0, for all x, and E phi(1)(X) = 0, we consider U-statistics of type [GRAPHICS] It is known that the conditions E phi(2)(X, X-1) < infinity and E phi(1)(2)(X) < infinity imply that the distribution function of T, say F, has a limit, say F-0, which can be described in terms of the eigenvalues of the Hilbert-Schmidt operator associated with the kernel phi(x, y). Under optimal moment conditions, we prove that [GRAPHICS] provided that at least nine eigenvalues of the operator do not vanish. Here F-1 denotes an Edgeworth-type correction. We provide explicit bounds for Delta(N) and for the concentration functions of statistics of type T.
Stichworte
statistics; central limit theorem; second order efficiency; bounds; Berry-Esseen; convergence rates; Edgeworth expansions; symmetric; von Mises statistics; degenerate U-statistics; U-statistics
Erscheinungsjahr
1999
Zeitschriftentitel
ANNALS OF PROBABILITY
Band
27
Ausgabe
1
Seite(n)
454-521
ISSN
0091-1798
Page URI
https://pub.uni-bielefeld.de/record/1623221

### Zitieren

Bentkus V, Götze F. Optimal bounds in non-Gaussian limit theorems for U-statistics. ANNALS OF PROBABILITY. 1999;27(1):454-521.
Bentkus, V., & Götze, F. (1999). Optimal bounds in non-Gaussian limit theorems for U-statistics. ANNALS OF PROBABILITY, 27(1), 454-521. https://doi.org/10.1214/aop/1022677269
Bentkus, V., and Götze, F. (1999). Optimal bounds in non-Gaussian limit theorems for U-statistics. ANNALS OF PROBABILITY 27, 454-521.
Bentkus, V., & Götze, F., 1999. Optimal bounds in non-Gaussian limit theorems for U-statistics. ANNALS OF PROBABILITY, 27(1), p 454-521.
V. Bentkus and F. Götze, “Optimal bounds in non-Gaussian limit theorems for U-statistics”, ANNALS OF PROBABILITY, vol. 27, 1999, pp. 454-521.
Bentkus, V., Götze, F.: Optimal bounds in non-Gaussian limit theorems for U-statistics. ANNALS OF PROBABILITY. 27, 454-521 (1999).
Bentkus, V, and Götze, Friedrich. “Optimal bounds in non-Gaussian limit theorems for U-statistics”. ANNALS OF PROBABILITY 27.1 (1999): 454-521.

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