Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-02T21:13:39.803Z Has data issue: false hasContentIssue false
  • English
  • Français

Regenerative processes in the infinite mean cycle case

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  14 July 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 Str., 1113 Sofia, Bulgaria.
∗ Postal address: Department of Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria.
Get access Rights & Permissions [Opens in a new window]

Abstract

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

Keywords

alternating regenerative processesalternating renewal processesinfinite mean cycleslimiting distributions

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 165 - 179
Copyright
Copyright © by the 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.)

References

Anderson, K. K., and Athreya, K. B. (1987). A renewal theorem in the infinite mean case. Ann. Prob. 15, 388–393.CrossRefGoogle Scholar
Assmusen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Athreya, K. B., and Ney, P. (1978). A new approach to the limit theorems of recurrent Markov chains. Trans. Amer. Math. Soc. 245, 157–181.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Dobrushin, R. L. (1955). Lemma on the limit of compound random functions. Uspekhi Mat. Nauk. 10, 157–159 (in Russian).Google Scholar
Dynkin, E. B. (1961). Some limit theorems for sums of independent random variables with infinite mathematical expectations. In Selected Translations Math. Statist. and Prob., Vol. 1. Inst. Math. Statist. and Amer. Math. Soc., Providence, RI, pp. 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. (1971). An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Kalashnikov, V., and Thorisson, H. (eds) (1994). Applications of coupling and regeneration. Acta Appl. Math. 34 (Special issue).Google Scholar
Kingman, J. F. C. (1972). Regenerative Phenomena. John Wiley, New York.Google Scholar
Lamperti, J. (1962). An invariance principle in renewal theory. Ann. Math. Statist. 33, 685–696.Google Scholar
Li, Z. H. (2000). Asymptotic behavior of continuous time and state branching processes. J. Austral. Math. Soc. A 68, 68–84.Google Scholar
Lindvall, T. (1992). Lectures on Coupling Method. John Wiley, New York.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. (1999). Limit theorems for alternating renewal processes. Submitted.Google Scholar
Mitov, K. V., Grishechkin, S. A., and Yanev, N. M. (1996). Limit theorems for regenerative processes. C. R. Acad. Bulgare Sci. 49, 29–32.Google Scholar
Mitov, K. V., Grishechkin, S. A., and Yanev, N. M. (1997). Two-stage renewal processes. C. R. Acad. Bulgare Sci. 50, 13–16.Google Scholar
Sigman, K., and Wolff, R. W. (1993). A review of regenerative processes. SIAM Rev. 35, 269–288.CrossRefGoogle Scholar
Smith, W. L. (1955). Regenerative stochastic processes. Proc. R. Soc. London A 232, 9–48.Google Scholar
Smith, W. L. (1958). Renewal theory and its ramifications. J. R. Statist. Soc. B 20, 6–31.Google Scholar
Teugels, J. L. (1968). Renewal theorems when the first or the second moment is infinite. Ann. Math. Statist. 39, 1210–1219.Google Scholar