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Continuity for multi-type branching processes with varying environments

Part of: Stochastic processes Markov 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
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

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