Short Communications

A nonuniform estimate for the error in short asymptotic expansions in Hilbert space

S. A. Bogatyrev

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: This work considers short asymptotic expansions of the probability for a sum of independent random elements to hit a ball in Hilbert space. An estimate for the error of the decomposition which is optimal with respect to a number of summands and depending only on at most 12 eigenvalues of the covariance operator of a summand is obtained. The error decreases if the distance between the bound of the ball and the zero element increases.

Keywords: short asymptotic expansions, Hilbert space, nonuniform estimate.

Received: 19.08.2002

DOI: 10.4213/tvp3780

 English version: Theory of Probability and its Applications, 2003, 47:4, 689–692

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