Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T12:26:58.910Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for alternating renewal processes in the infinite mean case

Part of:Special processes

Published online by Cambridge University Press:  19 February 2016

K. V. Mitov and
N. M. Yanev
K. V. Mitov*
Affiliation:
Bulgarian Academy of Sciences
N. M. Yanev*
Affiliation:
Bulgarian Academy of Sciences
*
∗ Postal address: Department of Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Street, 1113 Sofia, Bulgaria.
∗ Postal address: Department of Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Street, 1113 Sofia, Bulgaria.
Get accessRights & Permissions [Opens in a new window]

Abstract

The asymptotic behaviour of an occupation-time process associated with alternating renewal processes is investigated in the infinite mean cycle case. The limit theorems obtained extend some asymptotic results proved by Dynkin (1955), Lamperti (1958) and Erickson (1970) for the classical spent lifetime process. Some new phenomena are also presented.

Keywords

Alternating renewal processesstable lawsspent timelimiting distributions

MSC classification

Primary:60K05: Renewal theory

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001, pp. 896 - 911
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This paper is supported by NFSI-Bulgaria, Grant No. MM 704/97.

References

Anderson, K. K. and Athreya, K. B. (1987). A renewal theorem in the infinite mean case. Ann. Prob. 15, 388–393.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Dynkin, E. B. (1955). Some limit theorems for sums of independent random variables with infinite mathematical expectations. Izv. Akad. Nauk. SSSR Ser. Mat. 19, 247–266 (in Russian). English translation: Selected Translations Math. Statist. Prob. 1, 171–189.Google Scholar
Erickson, K. B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151, 263–291.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Kovalenko, I. N., Kuznetzov, N. U. and Shurenkov, V. M. (1983). Stochastic Processes Handbook. Naukova Dumka, Kiev (in Russian).Google Scholar
Lamperti, J. (1958). Some limit theorems for stochastic processes. J. Math. Mech. 7, 433–448.Google Scholar
Mitov, K. V. (1999). Limit theorems for regenerative excursion processes. Serdica Math. J. 25, 19–40.Google Scholar
Mitov, K. V. and Yanev, N. M. (1985). Bellman–Harris branching processes with state-dependent immigration. J. Appl. Prob. 22, 757–765.Google Scholar
Mitov, K. V. and Yanev, N. M. (1989). Bellman–Harris branching processes with a special type of state-dependent immigration. Adv. Appl. Prob. 21, 270–283.Google Scholar
Mitov, K. V. and Yanev, N. M. (2001). Regenerative processes in the infinite mean cycle case. J. Appl. Prob. 38, 165–179.Google Scholar
Sigman, K. and Wolff, R. W. (1993). A review of regenerative processes. SIAM Rev. 35, 269–288.Google Scholar
Teugels, J. L. (1968). Renewal theorems when the first or the second moment is infinite. Ann. Math. Statist. 39, 1210–1219.CrossRefGoogle Scholar