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Tightness for Maxima of Generalized Branching Random Walks

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  04 February 2016

Ming Fang
Ming Fang*
Affiliation:
University of Minnesota
*
∗ Current address: School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, China. Email address: fang0086@umn.edu
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Abstract

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We study generalized branching random walks on the real line R that allow time dependence and local dependence between siblings. Specifically, starting from one particle at time 0, the system evolves such that each particle lives for one unit amount of time, gives birth independently to a random number of offspring according to some branching law, and dies. The offspring from a single particle are assumed to move to new locations on R according to some joint displacement distribution; the branching laws and displacement distributions depend on time. At time n, Fn(·) is used to denote the distribution function of the position of the rightmost particle in generation n. Under appropriate tail assumptions on the branching laws and offspring displacement distributions, we prove that Fn(· - Med(Fn)) is tight in n, where Med(Fn) is the median of Fn. The main part of the argument is to demonstrate the exponential decay of the right tail 1 - Fn(· - Med(Fn)).

Keywords

Branching random walkrecursiontightness

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 652 - 670
Copyright
© Applied Probability Trust 

References

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