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Branching random walk in varying environments

Published online by Cambridge University Press:  01 July 2016

C. F. Klebaner
C. F. Klebaner*
Affiliation:
University of Melbourne
*
∗Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, VIC3052, Australia.
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Abstract

The branching random walk model is generalized towards generation-dependent displacement and reproduction distributions. Asymptotic theory of branching random walk in varying environments from the L2 point of view is given. If Zn(x) is the number of nth-generation particles to the left of x, then under appropriate conditions for suitably chosen xn, Zn (xn)/Zn (+∞) converges in L2 completely to a limiting distribution. Sufficient conditions for almost sure convergence are given. As a corollary an analogue of the central limit theorem for the proportion of particles of the nth generation in time interval In in the age-dependent Crump–Mode–Jagers process is obtained.

Keywords

POINT PROCESSESBRANCHING RANDOM WALKAGE-DEPENDENT BRANCHING PROCESSESMEAN SQUARE CONVERGENCECENTRAL LIMIT ANALOGUES
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 2 , June 1982 , pp. 359 - 367
Copyright
Copyright © Applied Probability Trust 1982 

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