Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T11:11:58.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Random environment branching processes with equal environmental extinction probabilities

Published online by Cambridge University Press:  14 July 2016

Donald C. Raffety
Donald C. Raffety*
Affiliation:
University of Texas
Get access Rights & Permissions [Opens in a new window]

Abstract

R-positivity theory for Markov chains is used to obtain results for random environment branching processes whose environment random variables are independent and identically distributed and whose environmental extinction probabilities are equal. For certain processes whose eventual extinction is almost sure, it is shown that the distribution of population size conditioned by non-extinction at time n tends to a left eigenvector of the transition matrix. Limiting values of other conditional probabilities are given in terms of this left eigenvector and it is shown that the probability of non-extinction at time n approaches zero geometrically as n approaches ∞. Analogous results are obtained for processes whose extinction is not almost sure.

Keywords

BRANCHING PROCESSESRANDOM ENVIRONMENTSR-POSITIVE MARKOV CHAINSGENERATING FUNCTIONSLIMIT THEOREMS
Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 659 - 665
Copyright
Copyright © Applied Probability Trust 1973 

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.)

Footnotes

*Now at St. Edwards University, Austin, Texas.

References

[1] Athreya, K. B. and Karlin, S. (1970) Branching processes with random environments. Bull. Amer. Math. Soc. 76, 865–870.Google Scholar
[2] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.10.1007/978-3-642-51866-9Google Scholar
[3] Heathcote, C. R., Seneta, E., and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Teor. Veroyat. Primen. 12, 341–346.Google Scholar
[4] Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.10.2307/3212128Google Scholar
[5] Smith, W. (1968) Necessary conditions for almost sure extinction of a branching process with random environment. Ann. Math. Statist. 39, 2136–2140.10.1214/aoms/1177698045Google Scholar
[6] Smith, W. and Wilkinson, W. (1969) On branching processes in random environments. Ann. Math. Statist. 40, 814–827.10.1214/aoms/1177697589Google Scholar
[7] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. 13, 7–28.Google Scholar
[8] Wilkinson, W. (1967) Branching processes in stochastic environments. Mimeo Series No. 544, Institute of Statistics, University of North Carolina.Google Scholar