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Teor. Veroyatnost. i Primenen., 1996, Volume 41, Issue 3, Pages 481–504 (Mi tvp3127)  

Normal approximation of $U$-statistics in Hilbert space

Yu. V. Borovskikha, M. L. Purib, V. V. Sazonovc

a Petersburg State Transport University
b Indiana University, Department of Mathematics, USA
c Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $\{U_n\}$, $n=1,2\ldots$ be Hilbert space $H$-valued $U$-statistics with kernel $\Phi(\cdotp,\cdot)$, corresponding to a sequence of observations (random variables) $X_1,X_2,\ldots $. The rate of convergence on balls in the central limit theorem for $\{U_n\}$ is investigated. The obtained estimate is of order $n^{-1/2}$ and depends explicitly on $\mathbb E\|\Phi(X_1,X_2)\|^3$ and on the trace and the first nine eigenvalues of the covariance operator of $\mathbb E(\Phi(X_1,X_2)|X_1)$.

Keywords: $U$-statistic, Hilbert space, central limit theorem, normal (Gaussian) approximation, rate of convergence.

DOI: https://doi.org/10.4213/tvp3127

Full text: PDF file (878 kB)

English version:
Theory of Probability and its Applications, 1997, 41:3, 405–424

Bibliographic databases:

Received: 17.05.1994

Citation: Yu. V. Borovskikh, M. L. Puri, V. V. Sazonov, “Normal approximation of $U$-statistics in Hilbert space”, Teor. Veroyatnost. i Primenen., 41:3 (1996), 481–504; Theory Probab. Appl., 41:3 (1997), 405–424

Citation in format AMSBIB
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\pages 481--504
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\jour Theory Probab. Appl.
\yr 1997
\vol 41
\issue 3
\pages 405--424
\crossref{https://doi.org/10.1137/S0040585X97975198}
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