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Critical branching random walks with small drift

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  • Zheng, Xinghua

Abstract

We study critical branching random walks (BRWs) U(n) on where the displacement of an offspring from its parent has drift towards the origin and reflection at the origin. We prove that for any [alpha]>1, conditional on survival to generation [n[alpha]], the maximal displacement is . We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like yn[alpha] for some y>0, [alpha]>1, and the particles are concentrated in , then the measure-valued processes associated with the BRWs converge to a measure-valued process, which, at any time t>0, distributes its mass over like an exponential distribution.

Suggested Citation

  • Zheng, Xinghua, 2010. "Critical branching random walks with small drift," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1821-1836, August.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:9:p:1821-1836
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    Cited by:

    1. Sun, Hongyan & Zhang, Lin, 2018. "Large deviation principle for the maximal positions in critical branching random walks with small drifts," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 31-39.

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