Hostname: page-component-5cf477f64f-tgq86 Total loading time: 0 Render date: 2025-04-04T22:56:45.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for decomposable branching processes in a random environment

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  30 March 2016

Vladimir Vatutin  and
Quansheng Liu
Vladimir Vatutin*
Affiliation:
Steklov Mathematical Institute
Quansheng Liu*
Affiliation:
Université de Bretagne-Sud
*
∗ Postal address: Steklov Mathematical Institute, Gubkin str. 8, 119991, Moscow, Russia. Email address: [email protected]
∗∗ Postal address: Université de Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France.
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.

We study the asymptotics of the survival probability for the critical and decomposable branching processes in a random environment and prove Yaglom-type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in a constant environment.

Keywords

Decomposable branching processessurvival probabilityrandom environment

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F17: Functional limit theorems; invariance principles 60J85: Applications of branching processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 877 - 893
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Afanasyev, V. I. (1999). On the maximum of a critical branching process in a random environment. Discrete Math. Appl. 9 267-284.Google Scholar
Afanasyev, V. I., Geiger, J., Kersting, G. and Vatutin, V. A. (2005). Criticality for branching processes in random environment. Ann. Prob. 33 645-673.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Foster, J. and Ney, P. (1976). Decomposable critical multi-type branching processes. Sankhyā A 38 28-37.Google Scholar
Foster, J. and Ney, P. (1978). Limit laws for decomposable critical branching processes. Z. Wahrscheinlichkeitsth. 46 13-43.CrossRefGoogle Scholar
Geiger, J. and Kersting, G. (2001). The survival probability of a critical branching process in a random environment. Theory Prob. Appl. 45 517-525.Google Scholar
Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40 814-827.CrossRefGoogle Scholar
Vatutin, V., Dyakonova, E., Jagers, P. and Sagitov, S. (2012). A decomposable branching process in a Markovian environment. Internat. J. Stoch. Anal. 2012 694285.Google Scholar