Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T07:30:19.713Z Has data issue: false hasContentIssue false
  • English
  • Français

Explosive Markov branching processes: entrance laws and limiting behaviour

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
* Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The supercritical Markov branching process is examined in the case where the minimal version of the process has strictly substochastic transition laws. This provides a nice example of the general construction theory for discrete-state Markov processes.

Entrance laws corresponding to the minimal process are characterised. Limit properties of the processes constructed from these entrance laws are examined. All such processes which are honest and cannot hit zero are ergodic. Otherwise these processes are λ-positive and limit theorems conditional on not having left the positive states are given.

A connection is made with recent work on the general construction problem when a λ-subinvariant measure is given. The case where immigration is allowed is mentioned.

Keywords

MARKOV PROCESSESENTRANCE LAWSINVARIANT MEASURES

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 737 - 756
Copyright
Copyright © Applied Probability Trust 1993 

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, W. J. (1991) Continuous-Time Markov Chains. Springer-Verlag, New York.Google Scholar
Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Chen, A. Y. and Renshaw, E. (1993) Recurrence of Markov branching processes with immigration. Stoch. Proc. Appl. 45, 231–242.Google Scholar
Chung, K. L. (1970) Lectures on Boundary Theory for Markov Chains. Princeton University Press, Princeton.Google Scholar
Doney, R. A. (1984) A note on some results of Schuh. J. Appl. Prob. 21, 192–196.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Holgate, P. (1967) Divergent population processes and mammal outbreaks. J. Appl. Prob. 4, 1–8.CrossRefGoogle Scholar
Hou, Zhenting and Qingfeng, Guo (1988) Homogeneous Denumerable Markov Processes. Springer-Verlag, Berlin.Google Scholar
Lamperti, J. (1967) Limiting distributions for branching processes. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2(2), 225–241.Google Scholar
Nair, M. G. and Pollett, P. K. (1993) On the relationship between µ-invariant measures and quasistationary distributions for continuous-time Markov chains. Adv. Appl. Prob. 25, 82–102. Correction: 25, 717-719.Google Scholar
Neveu, J. (1961) Lattice methods and submarkovian processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. II(2), 347–391.Google Scholar
Pakes, A. G. (1975) On Markov branching processes with immigration. Sankhya A 37, 129–138.Google Scholar
Pakes, A. G. (1989) Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements. Adv. Appl. Prob. 21, 243–269.CrossRefGoogle Scholar
Pollett, P. K. (1990a) A note on the classification of Q-processes when Q is not regular. J. Appl. Prob. 27, 278–290.Google Scholar
Pollett, P. K. (1990b) Invariant measures for Q-processes when Q is not regular. Adv. Appl. Prob. 23, 277–292.Google Scholar
Reuter, G. E. H. and Riley, P. W. (1972) The Feller property for Markov semigroups on a countable state space. J. London Math. Soc. (2) 5, 267–275.Google Scholar
Yang, Xiang-Qun (1990) The Construction Theory of Denumerable Markov Processes. Wiley, Chichester.Google Scholar