Representations and properties of weight functions in Tauberian theorems

Author:
B. M. Klykavka

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **77** (2007).

Journal:
Theor. Probability and Math. Statist. **77** (2008), 71-90

MSC (2000):
Primary 60G60, 62E20, 40E05

Published electronically:
January 16, 2009

MathSciNet review:
2432773

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We continue the studies of weight functions in Tauberian theorems for random fields. We obtain the rate of convergence of function series in the representation of a weight function and prove a recurrence relation for weight functions in spaces of various dimensions.

**1.**A. Ya. Olenko,*A Tauberian theorem for fields with the OR spectrum. II*, Teor. Ĭmovīr. Mat. Stat.**74**(2006), 81–97 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**74**(2007), 93–111. MR**2336781**, 10.1090/S0094-9000-07-00700-4**2.**Andriy Olenko,*Some properties of weight functions in Tauberian theorems. II*, Theory Stoch. Process.**13**(2007), no. 1-2, 194–204. MR**2343823****3.**N. N. Leonenko and A. V. Ivanov,*Statisticheskii analiz sluchainykh polei*, “Vishcha Shkola”, Kiev, 1986 (Russian). With a preface by A. V. Skorokhod. MR**917486****4.**M. Ĭ. Jadrenko,*Spektralnaya teoriya sluchainykh polei*, “Vishcha Shkola”, Kiev, 1980 (Russian). MR**590889****5.**Andriy Olenko and Boris Klykavka,*Some properties of weight functions in Tauberian theorems. I*, Theory Stoch. Process.**12**(2006), no. 3-4, 123–136. MR**2316570****6.**A. Ya. Olenko,*Upper bound on √𝑥𝐽ᵥ(𝑥) and its applications*, Integral Transforms Spec. Funct.**17**(2006), no. 6, 455–467. MR**2238583**, 10.1080/10652460600643445**7.**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****8.**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****9.**A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,*Tables of integral transforms. Vol. I*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR**0061695**

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Additional Information

**B. M. Klykavka**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
bklykavka@yahoo.com

DOI:
https://doi.org/10.1090/S0094-9000-09-00748-0

Keywords:
Tauberian theorems,
random fields,
covariance function,
spectral function,
weight function,
rate of convergence

Received by editor(s):
December 25, 2006

Published electronically:
January 16, 2009

Article copyright:
© Copyright 2009
American Mathematical Society