On some properties of asymptotic quasi-inverse functions

Buldygin, V.; Klesov, O.; Steinebach, J.

doi:10.1090/S0094-9000-09-00744-3

On some properties of asymptotic quasi-inverse functions

Authors: V. V. Buldygin, O. I. Klesov and J. G. Steinebach
Translated by: The authors
Journal: Theor. Probability and Math. Statist. 77 (2008), 15-30
MSC (2000): Primary 26A12; Secondary 26A48
DOI: https://doi.org/10.1090/S0094-9000-09-00744-3
Published electronically: January 14, 2009
MathSciNet review: 2432769
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A characterization of normalizing functions connected with the limiting behavior of ratios of asymptotic quasi-inverse functions is discussed. For nondecreasing functions, conditions are obtained that are necessary and sufficient for their asymptotic quasi-inverse functions to belong to the class of (so-called) $O$-regularly varying functions or to some of its subclasses.

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, Über einen O-Inversionssatz, Bull. Int. Acad. Youg. Sci. 29–30 (1936), 107–117.

N. K. Bari and S. B. Stečkin, Best approximations and differential properties of two conjugate functions, Trudy Moskov. Mat. Obšč. 5 (1956), 483–522 (Russian). MR 0080797

Simeon M. Berman, Sojourns and extremes of a diffusion process on a fixed interval, Adv. in Appl. Probab. 14 (1982), no. 4, 811–832. MR 677558, DOI https://doi.org/10.2307/1427025

Simeon M. Berman, The tail of the convolution of densities and its application to a model of HIV-latency time, Ann. Appl. Probab. 2 (1992), no. 2, 481–502. MR 1161063

N. H. Bingham and Charles M. Goldie, Extensions of regular variation. I. Uniformity and quantifiers, Proc. London Math. Soc. (3) 44 (1982), no. 3, 473–496. MR 656246, DOI https://doi.org/10.1112/plms/s3-44.3.473

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, Properties of a subclass of Avakumović functions and their generalized inverses, Ukraïn. Mat. Zh. 54 (2002), no. 2, 149–169 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 2, 179–206. MR 1952816, DOI https://doi.org/10.1023/A%3A1020178327423

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

V. V. Buldigīn, O. Ī. Klesov, and Ĭ. G. Shtaĭnebakh, On some properties of asymptotically quasi-inverse functions and their application. I, Teor. Ĭmovīr. Mat. Stat. 70 (2004), 9–25 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 70 (2005), 11–28. MR 2109819, DOI https://doi.org/10.1090/S0094-9000-05-00627-7

V. V. Buldigīn, O. Ī. Klesov, and Ĭ. G. Shtaĭnebakh, On some properties of asymptotically quasi-inverse functions and their application. II, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 34–48 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 71 (2005), 37–52. MR 2144319, DOI https://doi.org/10.1090/S0094-9000-05-00646-0

V. V. Buldigīn, O. Ī. Klesov, and Ĭ. G. Shtaĭnebakh, The PRV property of functions and the asymptotic behavior of solutions of stochastic differential equations, Teor. Ĭmovīr. Mat. Stat. 72 (2005), 10–23 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 72 (2006), 11–25. MR 2168132, DOI https://doi.org/10.1090/S0094-9000-06-00660-0

V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On some extensions of Karamata’s theory and their applications, Publ. Inst. Math. (Beograd) (N.S.) 80(94) (2006), 59–96. MR 2281907, DOI https://doi.org/10.2298/PIM0694059B

Daren B. H. Cline, Intermediate regular and $\Pi $ variation, Proc. London Math. Soc. (3) 68 (1994), no. 3, 594–616. MR 1262310, DOI https://doi.org/10.1112/plms/s3-68.3.594

Dragan Djurčić, $\scr O$-regularly varying functions and strong asymptotic equivalence, J. Math. Anal. Appl. 220 (1998), no. 2, 451–461. MR 1614959, DOI https://doi.org/10.1006/jmaa.1997.5807

Dragan Djurčić and Aleksandar Torgašev, Strong asymptotic equivalence and inversion of functions in the class $K_c$, J. Math. Anal. Appl. 255 (2001), no. 2, 383–390. MR 1815787, DOI https://doi.org/10.1006/jmaa.2000.7083

D. Drasin and E. Seneta, A generalization of slowly varying functions, Proc. Amer. Math. Soc. 96 (1986), no. 3, 470–472. MR 822442, DOI https://doi.org/10.1090/S0002-9939-1986-0822442-5

William Feller, One-sided analogues of Karamata’s regular variation, Enseign. Math. (2) 15 (1969), 107–121. MR 254905

I. V. Grinevich and Yu. S. Khokhlov, Domains of attraction of semistable laws, Teor. Veroyatnost. i Primenen. 40 (1995), no. 2, 417–422 (Russian, with Russian summary); English transl., Theory Probab. Appl. 40 (1995), no. 2, 361–366 (1996). MR 1346477, DOI https://doi.org/10.1137/1140038

Allan Gut, Oleg Klesov, and Josef Steinebach, Equivalences in strong limit theorems for renewal counting processes, Statist. Probab. Lett. 35 (1997), no. 4, 381–394. MR 1483025, DOI https://doi.org/10.1016/S0167-7152%2897%2900036-9

L. de Haan and U. Stadtmüller, Dominated variation and related concepts and Tauberian theorems for Laplace transforms, J. Math. Anal. Appl. 108 (1985), no. 2, 344–365. MR 793651, DOI https://doi.org/10.1016/0022-247X%2885%2990030-7

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

J. Karamata, Bemerkung über die vorstehende Arbeit des Herrn Avakumović, mit näherer Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen, Bull. Int. Acad. Youg. Sci. 29–30 (1936), 117–123.

Oleg Klesov, Zdzisław Rychlik, and Josef Steinebach, Strong limit theorems for general renewal processes, Probab. Math. Statist. 21 (2001), no. 2, Acta Univ. Wratislav. No. 2328, 329–349. MR 1911442

B. I. Korenblyum, On the asymptotic behavior of Laplace integrals near the boundary of a region of convergence, Dokl. Akad. Nauk SSSR (N.S.) 104 (1955), 173–176. MR 0074550

W. Matuszewska, On a generalization of regularly increasing functions, Studia Math. 24 (1964), 271–279. MR 167574, DOI https://doi.org/10.4064/sm-24-3-271-279

W. Matuszewska and W. Orlicz, On some classes of functions with regard to their orders of growth, Studia Math. 26 (1965), 11–24. MR 190273, DOI https://doi.org/10.4064/sm-26-1-11-24

Sidney I. Resnick, Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4, Springer-Verlag, New York, 1987. MR 900810

B. A. Rogozin, A Tauberian theorem for increasing functions of dominated variation, Sibirsk. Mat. Zh. 43 (2002), no. 2, 442–445, iii (Russian, with Russian summary); English transl., Siberian Math. J. 43 (2002), no. 2, 353–356. MR 1902831, DOI https://doi.org/10.1023/A%3A1014757424289

Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936

U. Stadtmüller and R. Trautner, Tauberian theorems for Laplace transforms, J. Reine Angew. Math. 311(312) (1979), 283–290. MR 549970, DOI https://doi.org/10.1016/0022-247X%2882%2990260-8

U. Stadtmüller and R. Trautner, Tauberian theorems for Laplace transforms in dimension $D>1$, J. Reine Angew. Math. 323 (1981), 127–138. MR 611447, DOI https://doi.org/10.1515/crll.1981.328.72

A. L. Yakymiv, Asymptotic properties of state change points in a random record process, Teor. Veroyatnost. i Primenen. 31 (1986), no. 3, 577–581 (Russian). MR 866880

A. L. Yakymiv, Asymptotics of the probability of nonextinction of critical Bellman-Harris branching processes, Trudy Mat. Inst. Steklov. 177 (1986), 177–205, 209 (Russian). Proc. Steklov Inst. Math. 1988, no. 4, 189–217; Probabilistic problems of discrete mathematics. MR 840684