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Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Alexander Iksanov  and
Matthias Meiners
Alexander Iksanov*
Affiliation:
National Taras Shevchenko University of Kyiv
Matthias Meiners*
Affiliation:
Universität Münster
*
∗Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kyiv, 01033 Kiev, Ukraine. Email address: iksan@unicyb.kiev.ua
∗∗Current address: Department of Mathematics, Uppsala University, 75106 Uppsala, Sweden.
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Abstract

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We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

Keywords

Branching random walkmartingalerate of convergencerenewal theory

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K05: Renewal theory 60G42: Martingales with discrete parameter
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 513 - 525
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

This work was commenced while A. Iksanov was visiting Münster in January 2009. A Iksanov thanks G. Alsmeyer, M. Meiners, and Institut für Mathematische Statistik for Invitation, hospitality, and financial support.

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