Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T12:46:51.440Z Has data issue: false hasContentIssue false
  • English
  • Français

Extinction and explosion of nonlinear Markov branching processes

Part of: Markov processes

Published online by Cambridge University Press:  09 April 2009

Anthony G. Pakes
Anthony G. Pakes
Affiliation:
School of Mathematics and StatisticsUniversity of Western Australia35 Stirling HighwayCrawley WA 6009Australiapakes@maths.uwa.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

Keywords

Markov branching processregular Markov processpopulation processexplosioninvariant measure and distribution.

MSC classification

Secondary: 60J75: Jump processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 3 , June 2007 , pp. 403 - 428
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Anderson, W. J., Continuous-time Markov chains (Springer-Verlag, New York. 1991).CrossRefGoogle Scholar
[2]Breiman, L., ‘On transient Markov chains with application to the uniqueness problem for Markov processes’, Ann. Math. Statist. 28 (1957), 499–503.CrossRefGoogle Scholar
[3]de Bruijn, N. G. and Erdös, P., ‘On a recursion formula and on some Tauberian theorems’, J. Research. Nat. Bur. Standards 50 (1953), 161–164.CrossRefGoogle Scholar
[4]Chen, A., ‘Ergodicity and stability of generalised Markov branching processes with resurrection’, J. Appl. Prob. 39 (2002), 786–803.CrossRefGoogle Scholar
[5]Chen, A., ‘Uniqueness and extinction properties of generalized Markov branching processes’, J. Math. Anal. Appl. 274 (2002), 482–494.CrossRefGoogle Scholar
[6]Chen, A., Li, J. and Ramesh, N. I., ‘General Harris regularity criterion for non-linear Markov branching processes’, Statist. Probab. Letters 76 (2005), 446–552.CrossRefGoogle Scholar
[7]Chen, A., ‘Uniqueness and extinction of weighted Markov branching processes’, Methodol. Comput. Appl. Prob. 7 (2005), 489–516.CrossRefGoogle Scholar
[8]Chen, A., Pollett, P., Li, J. and Zhang, H., ‘A remark on the uniqueness of weighted Markov branching processes’, J. Appl. Prob. 44 (2007), 274–283.CrossRefGoogle Scholar
[9]Chen, R.-R., ‘An extended class of time-continuous branching processes’, J. Appl. Prob. 34(1997), 14–23.CrossRefGoogle Scholar
[10]Doney, R. A., ‘A note on some results of Schuh’, J. Appl. Prob. 21(1984), 192–196.CrossRefGoogle Scholar
[11]Erickson, K. B., ‘The strong law of large numbers when the mean is undefined’, Trans. Amer. Math. Soc. 185 (1973), 371–381.CrossRefGoogle Scholar
[12]Etheridge, A. M., An introduction to superprocesses (American Mathematical Society, Providence, RI, 2000).CrossRefGoogle Scholar
[13]Garsia, A. M.. Orey, S. and Rodemich, E., ‘Asymptotic behaviour of successive coefficients of some power series’, Illinois J. Math. 6 (1962), 620–629.CrossRefGoogle Scholar
[14]Hamza, K. and Klebaner, F. C., ‘Conditions for integrability of Markov chains’, J. Appl. Probab. 32 (1995), 541–547.CrossRefGoogle Scholar
[15]Harris, T. E.. The theory of branching processes (Springer-Verlag, Berlin, 1963).CrossRefGoogle Scholar
[16]Ikeda, N., Nagasawa, M. and Watanabe, S., ‘Branching Markov processes II’, J. Math. Kyoto Univ. 8 (1968), 365–410.Google Scholar
[17]Kersting, G. and Klebaner, F. C., ‘Sharp conditions for non explosions and explosions in Markov jump processes’, Ann. Prob. 23 (1995), 268–272.CrossRefGoogle Scholar
[18]Küster, P., ‘Generalized Markov branching processes with state-dependent offspring distributions’, Z. Wahrsch. Verw. Gebiete 64 (1983), 475–503.CrossRefGoogle Scholar
[19]Lenz, N., ‘Extinction probability, regularity and asymptotic growth of Markovian populations’, J. Appl. Prob. 18 (1981), 1–18.CrossRefGoogle ScholarPubMed
[20]Lipow, C., ‘Limiting diffusions for population-size dependent branching processes’, J. Appl. Prob. 14 (1977), 14–24.CrossRefGoogle Scholar
[21]Pakes, A. G., ‘Conditional limit theorems for a left-continuous random walk’, J. Appl. Prob. 10 (1973), 39–53.CrossRefGoogle Scholar
[22]Pakes, A. G., ‘The branching-catastrophe process’. Stoch. Processes. Appl. 23 (1986), 1–33.CrossRefGoogle Scholar
[23]Pakes, A. G., ‘Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements’, Adv. Appl. Prob. 21 (1989), 243–269.CrossRefGoogle Scholar
[24]Pakes, A. G., ‘Divergence rates for explosive birth processes’, Stoch. Processes Appl. 41 (1992), 91–99.CrossRefGoogle Scholar
[25]Pakes, A. G., ‘Explosive Markov branching processes: Entrance laws and limiting behaviour’, Adv. Appl. Prob. 25 (1993), 737–756.CrossRefGoogle Scholar
[26]Pakes, A. G. and Tavare, S., ‘Comments on the age distribution of Markov processes’. Adv. Appl. Prob. 13 (1981), 681–703.CrossRefGoogle Scholar
[27]Reinhard, I., ‘The qualitative behaviour of some slowly growing population-dependent Markov branching processes’, in: Stochastic Modelling in Biology (ed. Tautu, P.) (World Scientific, Singapore, 1990) pp. 267–277.Google Scholar
[28]Sagitov, S. M., ‘On an explosive branching process’, Theor. Prob. Appl. 40 (1996), 575–577.CrossRefGoogle Scholar
[29]Savits, T. H., ‘The explosion problem for branching Markov processes’, Osaka J. Math. 6 (1969), 375–395.Google Scholar
[30]Yang, Y. N., ‘Asymptotic properties of the stationary measure of a Markov branching process’, J. Appl. Prob. 10 (1973), 447–450.CrossRefGoogle Scholar
[31]Zolotarev, V. M., ‘More exact statements of several theorems in the theory of branching processes’, Theor. Prob. Appl. 2 (1957), 245–253.CrossRefGoogle Scholar