×

Asymptotic properties of absolutely continuous functions and strong laws of large numbers for renewal processes. (English. Ukrainian original) Zbl 1301.26006

Theory Probab. Math. Stat. 87, 1-12 (2013); translation from Teor. Jmovirn. Mat. Stat. 87, 1-11 (2012).
Summary: In this paper, strong laws of large numbers for renewal processes constructed from compound counting processes are studied. In particular, a strong law of large numbers is proved for renewal processes constructed from compound Poisson processes with absolutely continuous rate functions.

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26A48 Monotonic functions, generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Aljančić and D. AranÄ’elović, 0-regularly varying functions, Publ. Inst. Math. (Beograd) (N.S.) 22(36) (1977), 5 – 22.
[2] D. AranÄ’elović, \?-regular variation and uniform convergence, Proceedings of the Third Annual Meeting of the International Workshop in Analysis and its Applications, 1990, pp. 25 – 40. · Zbl 0731.26004
[3] V. G. Avakumović, Über einen \( O\)-Inversionssatz, Bull. Int. Acad. Youg. Sci. 29-30 (1936), 107-117. · JFM 62.0223.01
[4] 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).
[5] Simeon M. Berman, Sojourns and extremes of a diffusion process on a fixed interval, Adv. in Appl. Probab. 14 (1982), no. 4, 811 – 832. · Zbl 0494.60076 · doi:10.2307/1427025
[6] 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. · Zbl 0752.62014
[7] 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. , https://doi.org/10.1112/plms/s3-44.3.473 N. H. Bingham and Charles M. Goldie, Extensions of regular variation. II. Representations and indices, Proc. London Math. Soc. (3) 44 (1982), no. 3, 497 – 534. · Zbl 0486.26002 · doi:10.1112/plms/s3-44.3.497
[8] 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. · Zbl 0617.26001
[9] 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. · Zbl 1006.60087 · doi:10.1023/A:1020178327423
[10] 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.
[11] 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.
[12] 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.
[13] V. V. Buldigīn, O. Ī. Klesov, and Ĭ. G. Shtaĭnebakh, On some properties of asymptotically quasi-inverse functions, Teor. Ĭmovīr. Mat. Stat. 77 (2007), 13 – 27 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 77 (2008), 15 – 30.
[14] Daren B. H. Cline, Intermediate regular and \Pi variation, Proc. London Math. Soc. (3) 68 (1994), no. 3, 594 – 616. · Zbl 0793.26004 · doi:10.1112/plms/s3-68.3.594
[15] Dragan Djurčić, \?-regularly varying functions and strong asymptotic equivalence, J. Math. Anal. Appl. 220 (1998), no. 2, 451 – 461. · Zbl 0920.26004 · doi:10.1006/jmaa.1997.5807
[16] Dragan Djurčić and Aleksandar Torgašev, Strong asymptotic equivalence and inversion of functions in the class \?_{\?}, J. Math. Anal. Appl. 255 (2001), no. 2, 383 – 390. · Zbl 0991.26002 · doi:10.1006/jmaa.2000.7083
[17] William Feller, One-sided analogues of Karamata’s regular variation, Enseignement Math. (2) 15 (1969), 107 – 121. · Zbl 0177.08201
[18] Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. · Zbl 0242.60003
[19] 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. · Zbl 0581.44003 · doi:10.1016/0022-247X(85)90030-7
[20] J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53. · JFM 56.0907.01
[21] J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France 61 (1933), 55 – 62 (French). · Zbl 0008.00807
[22] 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. · Zbl 0015.25004
[23] 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. · Zbl 1001.60024
[24] 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.
[25] W. Matuszewska, On a generalization of regularly increasing functions, Studia Math. 24 (1964), 271 – 279. · Zbl 0171.33503
[26] W. Matuszewska and W. Orlicz, On some classes of functions with regard to their orders of growth, Studia Math. 26 (1965), 11 – 24. · Zbl 0134.31604
[27] 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. · Zbl 0633.60001
[28] Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. · Zbl 0324.26002
[29] U. Stadtmüller and R. Trautner, Tauberian theorems for Laplace transforms, J. Reine Angew. Math. 311/312 (1979), 283 – 290. · Zbl 0484.44001 · doi:10.1016/0022-247X(82)90260-8
[30] U. Stadtmüller and R. Trautner, Tauberian theorems for Laplace transforms in dimension \?>1, J. Reine Angew. Math. 323 (1981), 127 – 138. · Zbl 0457.44004 · doi:10.1515/crll.1981.328.72
[31] 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).
[32] 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. · Zbl 0607.60075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.