On the Maximal Displacement of a Critical Branching Random Walk

2012/12/12 by Steven P. Lalley, Lalley, Steven P., Yuan Shao +1
Mathematics · #60J80 (Primary) #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60J80

paper · pdf · doi:10.48550/arxiv.1212.2933

corrected error in proof of Theorem 1

arxiv created 2014/03/28 · arxiv updated 2014/03/31

Abstract

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the integers. When the offspring distribution has mean 1 the branching process is critical, and therefore dies out with probability 1. We prove that if the particle jump distribution has mean zero, positive finite variance η², and finite 4+ε moment, and if the offspring distribution has positive variance σ² and finite third moment then the distribution of the rightmost position M reached by a particle of the branching random walk satisfies P\M ≥ x\∼ 6η²/ (σ²x²) as x → ∞. We also prove a conditional limit theorem for the distribution of the rightmost particle location at time n given that the process survives for n generations.

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