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 Electronic Journal of Probability > Vol. 9 (2004) > Paper 12 open journal systems 


Hierarchical Equilibria of Branching Populations

Donald A. Dawson, Carleton University
Luis G. Gorostiza, Centro de Investigacion y de Estudios Avanzados, Mexico D.F., Mexico
Anton Wakolbinger, Goethe Universitat, Frankfurt am Main, Germany


Abstract
Abstract. The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group $Omega_N$ consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit $Ntoinfty$ (called the {em hierarchical mean field limit}), the equilibrium aggregated populations in a nested sequence of balls $B^{(N)}_ell$ of hierarchical radius $ell$ converge to a backward Markov chain on $mathbb{R_+}$. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.


Full text: PDF

Pages: 316-381

Published on: April 26, 2004
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Electronic Journal of Probability. ISSN: 1083-6489