Abstract
We discuss the relationship between the strong asymptotic equivalence relation and the generalized inverse in the class
of all nondecreasing and unbounded functions, defined and positive on a half-axis [a, +∞) (a > 0). In the main theorem, we prove a proper characterization of the function class IRV∩
, where IRV is the class of all
-regularly varying functions (in the sense of Karamata) having continuous index function.
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References
Aljančić S., Arandjelović D., “
-Regularly varying functions,” Publ. Inst. Math. (Beograd), 22, 5–22 (1977).Bingham N. H., Goldie C. M., and Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge (1987).
Cline D. B. H., “Intermediate regular and Π-variation,” Proc. London Math. Soc., 68, No. 3, 594–616 (1994).
Djurčić D., “
-Regularly varying functions and strong asymptotic equivalence,” J. Math. Anal. Appl., 220, No. 2, 451–461 (1998).Berman S. M., “Sojourns and extremes of a diffusion process on a fixed interval,” Adv. in Appl. Probab., 14, No. 4, 811–832 (1982).
Berman S. M., “The tail of the convolution of densities and its application to a model of HIV-latency time,” Ann. Appl. Probab., 2, No. 2, 481–502 (1992).
Matuszewska W., “On a generalization of regularly increasing functions,” Studia Math., 24, 271–279 (1964).
Matuszewska W. and Orlicz W., “On some classes of functions with regard to their orders of growth,” Studia Math., 26, 11–24 (1965).
Korenblyum B. I., “On asymptotic behavior of Laplace integrals near the boundary of a convergence domain,” Dokl. Akad. Nauk SSSR, 104, No. 2, 173–176 (1955).
Stadtmüller U. and Trautner V., “Tauberian theorems for Laplace transforms,” J. Reine Angew. Math., 311/312, 283–290 (1979).
Stanojević Č. V., “
-Regularly varying convergence moduli of Fourier and Fourier-Stieltjes series,” Math. Ann., 279, 103–115 (1987).Seneta W., Functions of Regular Variation, Springer-Verlag, New York (1976) (Lecture Notes in Math.; 506).
Djurčić D. and Torgašev A., “Strong asymptotic equivalence and inversion of functions in the class K c ,” J. Math. Anal. Appl., 255, No. 2, 383–390 (2001).
de Haan L., On Regular Variation and Its Applications to the Weak Convergence of Sample Extremes, Math. Centrum, Amsterdam (1970) (Math. Centre Tract; 32).
Balkema A. A., Geluk J. L., and de Haan L., “An extension of Karamata’s Tauberian theorem and its connection with complementary convex functions,” Quart. J. Math. Oxford Ser., 30, No. 2, 385–416 (1979).
Arandjelović D., “
-Regular variation and uniform convergence,” Publ. Inst. Math. (Beograd), 48, 25–40 (1990).
-Regularly varying functions,” Publ. Inst. Math. (Beograd), 22, 5–22 (1977).
-Regularly varying functions and strong asymptotic equivalence,” J. Math. Anal. Appl., 220, No. 2, 451–461 (1998).
-Regularly varying convergence moduli of Fourier and Fourier-Stieltjes series,” Math. Ann., 279, 103–115 (1987).
-Regular variation and uniform convergence,” Publ. Inst. Math. (Beograd), 48, 25–40 (1990).Author information
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Original Russian Text Copyright © 2008 Djurčić D., Torgašev A., and Ješić S.
The authors were supported by the Ministry of Science of the Republic of Serbia (Grant 144031).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 4, pp. 786–795, July–August, 2008.
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Djurčić, D., Torgašev, A. & Ješić, S. The strong asymptotic equivalence and the generalized inverse. Sib Math J 49, 628–636 (2008). https://doi.org/10.1007/s11202-008-0059-z
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DOI: https://doi.org/10.1007/s11202-008-0059-z