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Continuity for multi-type branching processes with varying environments

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Owen Dafydd Jones
Owen Dafydd Jones*
Affiliation:
Australian National University
*
∗ Postal address: Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia. Email address: oj@maths.anu.edu.au.
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Abstract

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

Keywords

Branching processmulti-typevarying environmentcontinuity

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G17: Sample path properties

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 139 - 145
Copyright
Copyright © Applied Probability Trust 1999 

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References

Asmussen, S., and Hering, H. (1983). Branching Processes (Progr. Prob. 3) Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Cohn, H. (1996). On the asymptotic patterns of supercritical branching processes in varying environments. Ann. Appl. Prob. 6, 896–902.CrossRefGoogle Scholar
Cohn, H., and Jagers, P. (1994). General branching processes in varying environment. Ann. Appl. Prob. 4, 184–193.CrossRefGoogle Scholar
Cohn, H., and Nerman, O. (1990). On products of non-negative matrices. Ann. Prob. 18, 1806–1815.CrossRefGoogle Scholar
Grimmett, G. R., and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd edn. Clarendon Press, Oxford.Google Scholar
Jones, O. D. (1997). On the convergence of multi-type branching processes with varying environments. Ann. Appl. Prob. 7, 772–801.CrossRefGoogle Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.Google Scholar