A generalization of Karamata’s theorem on the asymptotic behavior of integrals
Authors:
V. V. Buldygin and V. V. Pavlenkov
Translated by:
O. Klesov
Journal:
Theor. Probability and Math. Statist. 81 (2010), 15-26
MSC (2010):
Primary 26A12, 26A48; Secondary 34C41
DOI:
https://doi.org/10.1090/S0094-9000-2010-00806-4
Published electronically:
January 14, 2011
MathSciNet review:
2667306
Full-text PDF Free Access
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Additional Information
Abstract: A generalization of Karamata’s theorem on the asymptotic behavior of integrals of regularly varying functions with oscillating components is obtained in the paper.
References
- S. Aljančić and D. Aranđelović, $0$-regularly varying functions, Publ. Inst. Math. (Beograd) (N.S.) 22(36) (1977), 5–22. MR 466438
- V. G. Avakumović, Uber einen O-Inversionssatz, Bull. Int. Acad. Youg. Sci. 1936, no. 29-30, 107–117.
- 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
- V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On factorization representations for Avakumović-Karamata functions with nondegenerate groups of regular points, Anal. Math. 30 (2004), no. 3, 161–192 (English, with English and Russian summaries). MR 2093756, DOI https://doi.org/10.1023/B%3AANAM.0000043309.79359.cc
- L. de Haan, On regular variation and its application to the weak convergence of sample extremes, Mathematical Centre Tracts, vol. 32, Mathematisch Centrum, Amsterdam, 1970. MR 0286156
- J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38–53.
- J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France 61 (1933), 55–62 (French). MR 1504998
- Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
References
- S. Aljančić and D. Arandelović, O-regularly varying functions, Publ. Inst. Math. (Beograd) (N.S.) 22(36) (1977), 5–22. MR 0466438 (57:6317)
- V. G. Avakumović, Uber einen O-Inversionssatz, Bull. Int. Acad. Youg. Sci. 1936, no. 29-30, 107–117.
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. MR 898871 (88i:26004)
- V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On factorization representations for Avakumović–Karamata functions with nondegenerate groups of regular points, Anal. Math. 30 (2004), 161–192. MR 2093756 (2005f:26029)
- L. de Haan, On Regular Variation and its Application to the Weak Convergence of Sample Extremes, Math. Centre Tracts, vol. 32, Amsterdam, 1975. MR 0286156 (44:3370)
- J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38–53.
- J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France 61 (1933), 55–62. MR 1504998
- E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976. MR 0453936 (56:12189)
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Additional Information
V. V. Buldygin
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (“KPI”), Peremogy Avenue 37, Kyiv 03056, Ukraine
Email:
matan@ntu-kpi.kiev.ua
V. V. Pavlenkov
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (“KPI”), Peremogy Avenue 37, Kyiv 03056, Ukraine
Keywords:
Regularly varying functions,
Karamata’s theorem,
asymptotic behavior of integrals,
oscillating functions
Received by editor(s):
November 3, 2009
Published electronically:
January 14, 2011
Article copyright:
© Copyright 2010
American Mathematical Society
