Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T19:43:32.221Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Survival Probability of a Branching Process in a Random Environment

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

J. C. D'souza  and
B. M. Hambly
J. C. D'souza*
Affiliation:
University of Aberdeen
B. M. Hambly*
Affiliation:
University of Edinburgh
*
∗ Postal address: Department of Mathematical Sciences, University of Aberdeen, Edward Wright Building, Dunbar Street, Aberdeen, AB24 3QY, UK.
∗∗ Postal address: Department of Mathematics and Statistics, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

Keywords

BRANCHING PROCESSESRANDOM ENVIRONMENTSVARYING ENVIRONMENTSGALTON-WATSON PROCESSSURVIVAL PROBABILITYRANDOM WALKSEXTINCTION TIMELARGE DEVIATIONS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 38 - 55
Copyright
Copyright © Applied Probability Trust 1997 

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

[1] Athreya, K. B. and Karlin, S. (1972) On branching processes in random environments: I. Extinction probability. Ann. Math. Statist. 42, 1499–1520.CrossRefGoogle Scholar
[2] Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[3] Borovkov, A. A. (1967) Boundary-value problems for random walks and large deviations in function spaces. Theory Prob. Appl. 12, 575–595.CrossRefGoogle Scholar
[4] Dekking, F. M. (1988) On the survival probability of a branching process in a finite state i.i.d. environment. Stoch. Proc. Appl. 27, 151–157.CrossRefGoogle Scholar
[5] Dembo, A. and Zajic, T. (1995) Large deviations: From empirical mean and measure to partial sum process. Stoch. Proc. Appl. 57, 191–224.CrossRefGoogle Scholar
[6] Dembo, A. and Zeitouni, O. (1993) Large Deviations Techniques and Applications. Jones and Bartlett, New York.Google Scholar
[7] D'Souza, J. C. (1995) The extinction time of the inhomogeneous branching process. In Branching Processes: Proc. First World Congress (Lecture Notes in Statistics 99). ed. Heyde, C. C.. Springer, Berlin. pp. 106–117.CrossRefGoogle Scholar
[8] D'Souza, J. C. and Biggins, J. D. (1992) The supercritical Galton-Watson process in varying environments. Stoch. Proc. Appl. 42, 39–47.CrossRefGoogle Scholar
[9] Fearn, D. H. (1971) Galton-Watson processes with generation dependence. Proc. 6th Berkeley Symp. Math. Statist. Prob. 4, 159–172.Google Scholar
[10] Fujimagari, T. (1980) On the extinction time distribution of a branching process in varying environments. Adv. Appl. Prob. 12, 350–366.CrossRefGoogle Scholar
[11] Goettge, R. T. (1976) Limit theorems for the supercritical Galton-Watson process in varying environments. Math. Biosci. 28, 171–190.CrossRefGoogle Scholar
[12] Kozlov, M. V. (1976) On the asymptotic behaviour of the probability of non-extinction for critical branching processes in a random environment. Theory Prob. Appl. 21, 791–804.CrossRefGoogle Scholar
[13] Liu, Q. (1996) On the survival probability of a branching process in a random environment. Ann. Inst. Henri Poincaré Prob. et Statist. 32, 10.Google Scholar
[14] Mogul'Skii, A. A. (1976) Large deviations for trajectories of multi-dimensional random walks. Theory Prob. Appl. 21, 300–315.CrossRefGoogle Scholar
[15] Rockafellar, R. T. (1970) Convex Analysis. Princeton University Press, Princeton, NJ.CrossRefGoogle Scholar
[16] Tanny, D. (1977) Limit theorems for branching processes in a random environment. Ann. Prob. 5, 100–116.CrossRefGoogle Scholar