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This article is cited in 1 scientific paper (total in 1 paper)
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.
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
Linking options:
https://www.mathnet.ru/eng/tvp3127https://doi.org/10.4213/tvp3127 https://www.mathnet.ru/eng/tvp/v41/i3/p481
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