Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Tauberian theorem for fields with an $ OR$ spectrum. II

Author: A. Ya. Olenko
Translated by: V. V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 93-111
MSC (2000): Primary 60G60, 62E20, 40E05; Secondary 60F05, 26A12, 44A15
Published electronically: June 29, 2007
MathSciNet review: 2336781
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Abstract: We consider homogeneous isotropic random fields whose spectra have some local singular properties. We prove Abelian and Tauberian theorems linking the local behavior of the spectral function and that of weighted integral functionals of random fields. Representations of weight functions in the form of the Hankel transform and series of functions are obtained. The asymptotic behavior is described in terms of functions of the class $ OR$. Some examples are given.

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Additional Information

A. Ya. Olenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01033, Ukraine

Keywords: Tauberian theorem, Abelian theorem, slowly varying functions, $OR$ class of functions, random fields, homogeneous fields, isotropic fields, functionals of a random field, spectral function, correlation function, asymptotics, strong dependence, Hankel transform, Bessel functions
Received by editor(s): February 1, 2005
Published electronically: June 29, 2007
Article copyright: © Copyright 2007 American Mathematical Society