Abelian and Tauberian theorems for random fields on two-point homogeneous spaces
Author:
A. A. Malyarenko
Translated by:
The author
Journal:
Theor. Probability and Math. Statist. 69 (2004), 115-127
MSC (2000):
Primary 60G60, 60G10; Secondary 40E05
DOI:
https://doi.org/10.1090/S0094-9000-05-00619-8
Published electronically:
February 8, 2005
MathSciNet review:
2110910
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider centered mean-square continuous random fields for which the variance of increments between two points depends only on the distance between these points. Relations between the asymptotic behavior of the variance of increments near zero and the asymptotic behavior of the spectral measure of the field near infinity are investigated. We prove several Abelian and Tauberian theorems in terms of slowly varying functions.
References
- 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
- V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716
- N. Ya. Vilenkin, Spetsial′nye funktsii i teoriya predstavleniĭ grupp, 2nd ed., “Nauka”, Moscow, 1991 (Russian, with Russian summary). MR 1177172
- Joseph A. Wolf, Spaces of constant curvature, 5th ed., Publish or Perish, Inc., Houston, TX, 1984. MR 928600
- N. N. Leonenko and A. Ya. Olenko, Tauberian and Abelian theorems for the correlation function of a homogeneous isotropic random field, Ukrain. Mat. Zh. 43 (1991), no. 12, 1652–1664 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 43 (1991), no. 12, 1539–1548 (1992). MR 1172306, DOI https://doi.org/10.1007/BF01066693
- A. A. Malyarenko, Local properties of Gaussian random fields on compact symmetric spaces, and Jackson-type and Bernstein-type theorems, Ukraïn. Mat. Zh. 51 (1999), no. 1, 60–68 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 51 (1999), no. 1, 66–75. MR 1712757, DOI https://doi.org/10.1007/BF02591915
- G. M. Molčan, Homogeneous random fields on symmetric spaces of rank one, Teor. Veroyatnost. i Mat. Statist. 21 (1979), 123–148, 167 (Russian, with English summary). MR 550252
- 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, DOI https://doi.org/10.1007/BF02378535
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517
- A. M. Yaglom, Certain types of random fields in $n$-dimensional space similar to stationary stochastic processes, Teor. Veroyatnost. i Primenen 2 (1957), 292–338 (Russian, with English summary). MR 0094844
- M. Ĭ. Jadrenko, Spektral′naya teoriya sluchaĭ nykh poleĭ, “Vishcha Shkola”, Kiev, 1980 (Russian). MR 590889
- Robert J. Adler, An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. MR 1088478
- R. Askey and N. H. Bingham, Gaussian processes on compact symmetric spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 2, 127–143. MR 423000, DOI https://doi.org/10.1007/BF00536776
- N. H. Bingham, Tauberian theorems for Jacobi series, Proc. London Math. Soc. (3) 36 (1978), no. 2, 285–309. MR 620813, DOI https://doi.org/10.1112/plms/s3-36.2.285
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871
- Mogens Flensted-Jensen and Tom H. Koornwinder, Positive definite spherical functions on a noncompact, rank one symmetric space, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976–1978), II, Lecture Notes in Math., vol. 739, Springer, Berlin, 1979, pp. 249–282. MR 560841
- Ramesh Gangolli, Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters, Ann. Inst. H. Poincaré Sect. B (N.S.) 3 (1967), 121–226. MR 0215331
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454
- 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, DOI https://doi.org/10.1515/rose.1993.1.1.57
- E. J. G. Pitman, On the behavior of the characteristic function of a probability distribution in the neighborhood of the origin, J. Austral. Math. Soc. 8 (1968), 423–443. MR 0231423
- A. M. Yaglom, Second-order homogeneous random fields, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 593–622. MR 0146880
References
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. II, McGraw–Hill, New York, 1953. MR 0058756 (15:419i)
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1998; English transl., Translations of Mathematical Monographs, vol. 188, Amer. Math. Soc., Providence, RI, 2000. MR 1743716 (2001g:60089)
- N. Ya. Vilenkin, Special Functions and the Theory of Group Representations, “Nauka”, Moscow, 1991; English transl., Translations of Mathematical Monographs, vol. 22, Amer. Math. Soc., Providence, RI, 1968. MR 1177172 (93d:33013)
- J. A. Wolf, Spaces of Constant Curvature, Publish or Perish, Wilmington, DE, 1984. MR 0928600 (88k:53002)
- N. N. Leonenko and A. Ya. Olenko, Tauberian and Abelian theorems for the correlation function of a homogeneous isotropic random field, Ukrain. Mat. Zh. 43 (1991), no. 12, 1652–1664; English transl. in Ukrainian Math. J. 43 (1991), 1539–1548. MR 1172306 (93f:60069)
- A. A. Malyarenko, Local properties of Gaussian random fields on compact symmetric spaces, and Jackson-type and Bernstein-type theorems, Ukrain. Mat. Zh. 51 (1999), no. 1, 60–68; English transl. in Ukrainian Math. J. 43 (1991), 66–75. MR 1712757 (2000j:60053)
- G. M. Molchan, Homogeneous random fields on symmetric spaces of rank one, Teor. Veroyatnost. Matem. Statist. (1979), no. 21, 123–148; English transl. in Theory Probab. Math. Statist. 21 (1980), 143–168. MR 0550252 (81f:60077)
- A. Ya. Olenko, Tauberian and Abelian theorems for random fields with strong dependence, Ukrain. Mat. Zh. 48 (1996), no. 3, 368–382; English transl. in Ukrainian Math. J. 48 (1996), 412–427. MR 1408658 (97k:60143)
- G. Szegő, Orthogonal Polynomials, Colloquium Publications, vol. XXIII, Amer. Math. Soc., Providence, RI, 1975. MR 0372517 (51:8724)
- A. M. Yaglom, Certain types of random fields in $n$-dimensional space similar to stationary stochastic processes, Teor. Veroyatnost. i Primenen. 2 (1957), 292–338; English transl. in Theory Probab. Appl. MR 0094844 (20:1353)
- M. I. Yadrenko, Spectral Theory of Random Fields, “Vyshcha Shkola”, Kiev, 1980; English transl., Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1983. MR 0590889 (82e:60001)
- R. J. Adler, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. MR 1088478 (92g:60053)
- R. Askey and N. H. Bingham, Gaussian processes on compact symmetric spaces, Z. Wahrscheinlichkeitstheorie verw. Gebiete 37 (1976), no. 2, 127–143. MR 0423000 (54:10984)
- N. H. Bingham, Tauberian theorems for Jacobi series, Proc. London Math. Soc. (3) 36 (1978), no. 2, 285–309. MR 0620813 (58:29795)
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 0898871 (88i:26004)
- M. Flensted-Jensen and T. H. Koornwinder, Positive definite spherical functions on a noncompact, rank one symmetric space, Analyse Harmonique sur les Groupes de Lie II, Lect. Notes Math., vol. 739, Springer-Verlag, Berlin–Heidelberg–New York, 1979, pp. 249–282. MR 0560841 (81j:43015)
- R. Gangolli, Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters, Ann. Inst. H. Poincaré, Sect. B 3 (1967), 121–226. MR 0215331 (35:6172)
- S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, vol. 34, Amer. Math. Soc., Providence, RI, 2001. MR 1834454 (2002b:53081)
- 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)
- E. J. G. Pitman, On the behavior of the characteristic function of a probability distribution in the neighborhood of the origin, J. Austral. Math. Soc. 8 (1968), 423–443. MR 0231423 (37:6978)
- A. M. Yaglom, Second order homogeneous random fields, Proc. IV Berkeley Symp. Math. Stat. Probab., vol. 2, 1961, pp. 593–622. MR 0146880 (26:4399)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60G60,
60G10,
40E05
Retrieve articles in all journals
with MSC (2000):
60G60,
60G10,
40E05
Additional Information
A. A. Malyarenko
Affiliation:
International Mathematical Centre, National Academy of Sciences of Ukraine
Address at time of publication:
Mälardalen University, P. O. Box 883, SE–721 23 Västerås, Sweden
Email:
anatoliy.malyarenko@mdh.se
Keywords:
Random field,
Abelian theorem,
Tauberian theorem,
two-point homogeneous space
Received by editor(s):
January 3, 2003
Published electronically:
February 8, 2005
Additional Notes:
This work is supported in part by the Foundation for Knowledge and Competence Development.
Article copyright:
© Copyright 2005
American Mathematical Society