Normal Approximation of U-Statistics in Hilbert Space

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Date

1997

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Theory of Probability & Its Applications

Abstract

Let $\{U_n\}$, $n=1,2,...,$ 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 $E\|\Phi(X_1,X_2)\|^3$ and on the trace and the first nine eigenvalues of the covariance operator of $E(\Phi(X_1,X_2)|X_1)$.

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Publisher's, offprint version

Keywords

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

Citation

Puri, M. L. “Normal approximation of U-statistics in Hilbert spaces.” Translation by SIAM,Theory of Probability and its Applications (1997), Volume 41 Issue 3, 481–504. Co-authors: Yu.V. Borovskich and V.V. Sazonov.

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