Critical Branching Random Walks with Small Drift

2009/11/12 by Xinghua Zheng, Zheng, Xinghua
Mathematics · Physics and Astronomy · #60G57 #60J80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60G57 #msc:60J80

paper · pdf · doi:10.48550/arxiv.0911.2401

openalex publication_date 2009/11/12 · arxiv created 2010/04/26 · arxiv updated 2010/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study critical branching random walks (BRWs) U⁽ⁿ⁾ on~ℤ₊ where for each n, the displacement of an offspring from its parent has drift~2β/√(n) towards the origin and reflection at the origin. We prove that for any~α>1, conditional on survival to generation~[n^α], the maximal displacement is asymptotically equivalent to (α-1)/(4β)√(n)log n. We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like~yn^α for some y>0 and α>1, and the particles are concentrated in~[0,O(√(n))], then the measure-valued processes associated with the BRWs, under suitable scaling converge to a measure-valued process, which, at any time~t>0, distributes its mass over~ℝ₊ like an exponential distribution.

Citations

• A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations
• Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions

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