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

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

doi:10.1090/S0094-9000-05-00646-0

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

Authors: V. V. Buldygin, O. I. Klesov and J. G. Steinebach
Translated by: The authors
Journal: Theor. Probability and Math. Statist. 71 (2005), 37-52
MSC (2000): Primary 26A12; Secondary 60F15
DOI: https://doi.org/10.1090/S0094-9000-05-00646-0
Published electronically: December 28, 2005
MathSciNet review: 2144319
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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

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 asymptotic quasi-inverse functions, Theory Stoch. Process. 9 (2003), no. 1-2, 30–51. MR 2080011

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

William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403

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

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

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 \Avakumovic, mit näherer Betrachtung einer Klasse von Funktionen, welche bei den Inversionssätzen vorkommen, Bull. Int. Acad. Youg. Sci. 29–30 (1936), 117–123.

G. Keller, G. Kersting, and U. Rösler, On the asymptotic behaviour of solutions of stochastic differential equations, Z. Wahrsch. Verw. Gebiete 68 (1984), no. 2, 163–189. MR 767799, DOI https://doi.org/10.1007/BF00531776

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

H. S. A. Potter, The mean values of certain Dirichlet series, II, Proc. London Math. Soc. (2) 47 (1940), 1–19. MR 5141, DOI https://doi.org/10.1112/plms/s2-47.1.1

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, 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