Theory of Probability and Mathematical Statistics

Some properties of asymptotic quasi-inverse functions and their applications. II

Author(s): V. V. Buldygin; O. I. Klesov; J. G. Steinebach
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 71 (2004).
Journal: Theor. Probability and Math. Statist. No. 71 (2005), 37-52.
MSC (2000): Primary 26A12; Secondary 60F15
Posted: December 28, 2005
Retrieve article in: PDF DVI PostScript

Abstract: We continue to study properties of functions which are asymptotic (quasi-)inverse for PRV and POV functions. The equivalence of all quasi-inverses for POV functions is proved. Under appropriate conditions, we derive the limiting behaviour of the ratio of asymptotic quasi-inverse functions from the corresponding asymptotics of their original versions. Several applications of these general results to the asymptotic stability of a Cauchy problem, to the asymptotics of the solution of a stochastic differential equation, and to the limiting behavior of generalized renewal processes are also presented.

1.: S. Aljancic and D. Arandelovic, -regularly varying functions, Publ. Inst. Math. Nouvelle Série 22(36) (1977), 5-22. MR 0466438 (57:6317)
2.: V. G. Avakumovic, Über einen O-Inversionssatz, Bull. Int. Acad. Youg. Sci. 29-30 (1936), 107-117.
3.: N. K. Bari and S. B. Stechkin, Best approximation and differential properties of two conjugate functions, Trudy Mosk. Mat. Obsch. 5 (1956), 483-522. (Russian) MR 0080797 (18,303e)
4.: N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. MR 0898871 (88i:26004)
5.: V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, Properties of a subclass of Avakumovic functions and their generalized inverses, Ukrain. Math. J. 54 (2002), 179-205. MR 1952816 (2003i:60044)
6.: V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, On asymptotic quasi-inverse functions, Theory Stoch. Process. 9(25) (2003), 30-51. MR 2080011 (2005g:26002)
7.: V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, Some properties of asymptotic quasi-inverse functions and their applications I, Teor. Imovir. ta Matem. Statyst. 70 (2004), 9-25; English transl. in Theory Probab. Math. Statist. 70 (2005), 11-28. MR 2109819 (2005i:26005)
8.: W. Feller, An Introduction to Probability Theory and Its Applications, Second edition, vol. 2, John Wiley, New York, 1971. MR 0270403 (42:5292)
9.: A. Gut, O. Klesov, and J. Steinebach, Equivalences in strong limit theorems for renewal counting processes, Statist. Probab. Lett. 35 (1997), 381-394. MR 1483025 (98m:60043)
10.: D. Drasin and E. Seneta, A generalization of slowly varying functions, Proc. Amer. Math. Soc. 96 (1986), no. 3, 470-472. MR 0822442 (87d:26002)
11.: J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53.
12.: J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamenteaux, Bull. Soc. Math. France 61 (1933), 55-62. MR 1504998
13.: J. Karamata, Bemerkung über die vorstehende Arbeit des Herrn Avakumovic, mit näherer Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen, Bull. Int. Acad. Youg. Sci. 29-30 (1936), 117-123.
14.: G. Keller, G. Kersting, and U. Rösler, On the asymptotic behaviour of solutions of stochastic differential equations, Z. Wahrsch. Verw. Geb. 68 (1984), 163-184. MR 0767799 (86i:60153)
15.: O. Klesov, Z. Rychlik, and J. Steinebach, Strong limit theorems for general renewal processes, Theory Probab. Math. Statist. 21 (2001), 329-349. MR 1911442 (2003j:60120)
16.: B. H. Korenblyum, On the asymptotic behaviour of Laplace integrals near the boundary of a region of convergence, Dokl. Akad. Nauk. USSR (N.S.) 109 (1956), 173-176. MR 0074550 (17,605a)
17.: H. S. A. Potter, The mean value of a Dirichlet series II, Proc. London Math. Soc. 47 (1942), 1-19. MR 0005141 (3:107e)
18.: E. Seneta, Regularly Varying Functions, Springer-Verlag, Berlin, 1976. MR 0453936 (56:12189)
19.: U. Stadtmüller and R. Trautner, Tauberian theorems for Laplace transforms, J. Reine Angew. Math. 311/312 (1979), 283-290. MR 0549970 (81f:44006)
20.: U. Stadtmüller and R. Trautner, Tauberian theorems for Laplace transforms in dimension , J. Reine Angew. Math. 323 (1981), 127-138. MR 0611447 (82i:44001)
21.: A. L. Yakymiv, Asymptotics of the probability of nonextinction of critical Bellman-Harris branching processes, Proc. Steklov Inst. Math. 4 (1988), 189-217. MR 0840684 (88d:60221)

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 26A12, 60F15

V. V. Buldygin
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (KPI), Pr. Peremogy 37, Kyiv 03056, Ukraine
Email: valbuld@comsys.ntu-kpi.kiev.ua

O. I. Klesov
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (KPI), Pr. Peremogy 37, Kyiv 03056, Ukraine
Email: oleg@tbimc.freenet.kiev.ua

J. G. Steinebach
Affiliation: Universität zu Köln, Mathematisches Institut, Weyertal 86--90, D--50931 Köln, Germany
Email: jost@math.uni-koeln.de

DOI: 10.1090/S0094-9000-05-00646-0
PII: S 0094-9000(05)00646-0
Received by editor(s): 27/FEB/2004
Posted: December 28, 2005
Additional Notes: This work has partially been supported by Deutsche Forschungsgemeinschaft under DFG grants 436 UKR 113/41/0-2 and 436 UKR 113/68/0-1.
Copyright of article: Copyright 2005, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1547-7363(e) 0094-9000(p)

Previous number \| Table of contents \| Next number Recently posted articles \| Most recent number \| All numbers Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS