Geodesic
Normal approximation of $U$-statistics in Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 3, pp. 481-504

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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. 
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     author = {Yu. V. Borovskikh and M. L. Puri and V. V. Sazonov},
     title = {Normal approximation of $U$-statistics in {Hilbert} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {481--504},
     publisher = {mathdoc},
     volume = {41},
     number = {3},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1996_41_3_a0/}
}
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Yu. V. Borovskikh; M. L. Puri; V. V. Sazonov. Normal approximation of $U$-statistics in Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 3, pp. 481-504. http://geodesic.mathdoc.fr/item/TVP_1996_41_3_a0/