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Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

Part of: Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Markus Bachmann
Markus Bachmann*
Affiliation:
Purdue University
*
∗ Postal address: Neuhofstr. 17, 60318 Frankfurt, Germany. Email address: MengMarkus@aol.com
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Abstract

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

Keywords

Branching random walkminimal positionage-dependent branching processfirst birth timenonlinear integral operatortravelling waveextreme value distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 159 - 176
Copyright
Copyright © Applied Probability Trust 2000 

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