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Tauberian theorem for fields with an  spectrum. II
Author(s):
A.
Ya.
Olenko
Translated by:
V. V. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 74
(2006).
Journal:
Theor. Probability and Math. Statist.
No. 74
(2007),
93-111.
MSC (2000):
Primary 60G60, 62E20, 40E05;
Secondary 60F05, 26A12, 44A15
Posted:
June 29, 2007
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Additional information
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  . Some examples are given.
References:
-
- 1.
- A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. MR 0058756 (15:419i)
- 2.
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, 1989. MR 1015093 (90i:26003)
- 3.
- S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, 1949. MR 0031582 (11:173d)
- 4.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, 1944. MR 0010746 (6:64a)
- 5.
- G. Laue, Tauberian and Abelian theorems for characteristic functions, Teor. Veroyatnost. Matem. Statist. 37 (1987), 78-92; English transl. in Theor. Probability Math. Statist. 37 (1988), 89-104. MR 913912 (89a:60044)
- 6.
- A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields, ``Vyshcha shkola'', Kiev, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989. MR 917486 (89e:62125); MR 1009786 (90g:62235)
- 7.
- N. N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum, Kluwer Academic Publishers, Dordrecht-Boston-London, 1999. MR 1687092 (2000k:60102)
- 8.
- F. J. Narcowich and J. D. Ward, Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices, J. Approx. Theory 69 (1992), 84-109. MR 1154224 (93c:41005)
- 9.
- A. Ya. Olenko, Tauberian theorems for random fields with an
 spectrum. I, Teor. Imovirnost. Matem. Statist. 73 (2005), 120-133; English transl. in Theor. Probability Math. Statist. 73 (2006), 135-149. MR 2213848 - 10.
- M. I. Yadrenko, Spectral Theory of Random Fields, Optimization Software Inc., New York, 1983. MR 697386 (84f:60003)
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60G60, 62E20, 40E05,
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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
Email:
olenk@univ.kiev.ua
DOI:
10.1090/S0094-9000-07-00700-4
PII:
S 0094-9000(07)00700-4
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):
1/FEB/2005
Posted:
June 29, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
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