A limit theorem for random fields with a singularity in the spectrum
Authors:
A. Ya. Olenko and B. M. Klykavka
Translated by:
O. Klesov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 81 (2010).
Journal:
Theor. Probability and Math. Statist. 81 (2010), 147158
MSC (2010):
Primary 60G60; Secondary 60F17
Published electronically:
January 20, 2011
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Homogeneous isotropic random fields with singularities in spectra at nonzero frequencies are studied. This class of fields generalizes the case of random fields with long range dependence where the spectrum has a singularity at the origin. We obtain a limit theorem for integral weight functionals of the field. We also discuss the difference between this class and the long range dependence.
 1.
A.
V. Ivanov and N.
N. Leonenko, Statistical analysis of random fields,
Mathematics and its Applications (Soviet Series), vol. 28, Kluwer
Academic Publishers Group, Dordrecht, 1989. With a preface by A. V.
Skorokhod; Translated from the Russian by A. I. Kochubinskiĭ. MR 1009786
(90g:62235)
 2.
M.
Ĭ. Yadrenko, Spectral theory of random fields,
Translation Series in Mathematics and Engineering, Optimization Software,
Inc., Publications Division, New York, 1983. Translated from the Russian.
MR 697386
(84f:60003)
 3.
N.
N. Leonenko and A.
Ya. Olenko, Tauberian theorems for correlation functions and limit
theorems for spherical averages of random fields, Random Oper.
Stochastic Equations 1 (1993), no. 1, 57–67. MR 1254176
(95a:60068), 10.1515/rose.1993.1.1.57
 4.
A.
Ya. Olenko, Tauberian and Abelian theorems for random fields with
strong dependence, Ukraïn. Mat. Zh. 48 (1996),
no. 3, 368–382 (Ukrainian, with English and Ukrainian
summaries); English transl., Ukrainian Math. J. 48
(1996), no. 3, 412–427 (1997). MR 1408658
(97k:60143), 10.1007/BF02378535
 5.
Nikolai
Leonenko, Limit theorems for random fields with singular
spectrum, Mathematics and its Applications, vol. 465, Kluwer
Academic Publishers, Dordrecht, 1999. MR 1687092
(2000k:60102)
 6.
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 (2008i:60085), 10.1090/S0094900007007004
 7.
G.
N. Watson, A Treatise on the Theory of Bessel Functions,
Cambridge University Press, Cambridge, England; The Macmillan Company, New
York, 1944. MR
0010746 (6,64a)
 8.
Christian
Houdré, Linear Fourier and stochastic analysis, Probab.
Theory Related Fields 87 (1990), no. 2,
167–188. MR 1080488
(92e:60096), 10.1007/BF01198428
 9.
R.
L. Dobrushin, Gaussian and their subordinated selfsimilar random
generalized fields, Ann. Probab. 7 (1979),
no. 1, 1–28. MR 515810
(80e:60069)
 10.
Péter
Major, Multiple WienerItô integrals, Lecture Notes in
Mathematics, vol. 849, Springer, Berlin, 1981. With applications to
limit theorems. MR 611334
(82i:60099)
 1.
 N. N. Leonenko and A. V. Ivanov, Statistical Analysis of Random Fields, Kluwer Academic Publishers, DordrechtBostonLondon, 1989. MR 1009786 (90g:62235)
 2.
 M. I. Yadrenko, Spectral Theory of Random Fields, Optimization Software Inc., New York, 1983 (distributed by SpringerVerlag). MR 697386 (84f:60003)
 3.
 N. N. Leonenko and A. Ya. Olenko, Tauberian theorems for correlation functions and limit theorems for spherical averages of random fields, Random Oper. Stoch. Eqs. 1 (1993), no. 1, 5767. MR 1254176 (95a:60068)
 4.
 A. Ya. Olenko, Tauberian and Abelian theorems for random fields with strong dependence, Ukrain. Mat. Zh. 48 (1996), no. 3, 368382; English transl. in Ukrainian Math. J. 48 (1996), no. 3, 412427. MR 1408658 (97k:60143)
 5.
 N. N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum, Kluwer Academic Publishers, 1999. MR 1687092 (2000k:60102)
 6.
 A. Ya. Olenko, Tauberian theorems for random fields with OR asymptotics. II, Teor. Imovirnost. Matem. Statist. 74 (2006), 8197; English transl. in Theory Probab. Math. Statist. 74 (2007), 93111. MR 2336781 (2008i:60085)
 7.
 G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944. MR 0010746 (6:64a)
 8.
 C. Houdre, Linear Fourier and stochastic analysis, Probab. Theory and Related Fields 87 (1990), 167188. MR 1080488 (92e:60096)
 9.
 R. L. Dobrushin, Gaussian and their subordinated selfsimilar random generalized fields, The Annals of Probability 7 (1979), no. 1, 128. MR 515810 (80e:60069)
 10.
 P. Major, Multiple WienerItô Integrals, Lecture Notes in Math., vol. 849, Springer, New York, 1981. MR 611334 (82i:60099)
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Additional Information
A. Ya. Olenko
Affiliation:
Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
Email:
a.olenko@latrobe.edu.au
B. M. Klykavka
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
bklykavka@yahoo.com
DOI:
http://dx.doi.org/10.1090/S009490002011008162
Keywords:
Random fields,
limit theorem,
integral weight functionals,
spectral functions,
long range dependence
Received by editor(s):
August 31, 2009
Published electronically:
January 20, 2011
Additional Notes:
Supported by the Swedish Institute grant SI01424/2007
Article copyright:
© Copyright 2011
American Mathematical Society
