Estimates for the rate of strong approximation in Hilbert space

Details Estimates for the rate of strong approximation in Hilbert space Estimates for the rate of strong approximation in Hilbert space

2012-03-26

arxiv.org/abs/1203.5695v1

published in Siberian Mathematical Journal, 2011, v. 52, no. 4, 796-808

Mathematics

Probability

15 pages

Scientific paper

The aim of this paper is to investigate, which infinite dimensional consequences follow from the main results of recently published paper of the authors (2009) (see Theorems 2 and 3). We show that the finite dimensional Theorem 3 implies meaningful estimates for the rate of strong Gaussian approximation of sums of i.i.d. Hilbert space valued random vectors $\xi_j$ with finite moments $E |\xi_j|^\gamma$, $\gamma>2$. We show that the rate of approximation depends substantially on the rate of decay of the sequence of eigenvalues of the covariance operator of summands.

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