On the maximal displacement of near-critical branching random walks

• 成果类型: Article
• 署名作者: Neuman, Eyal; Zheng, Xinghua
• 署名单位: Imperial College London; Hong Kong University of Science & Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01042-8
• 发表日期: 2021
• 页码: 199-232
• 关键词: limit-theorems; martingale convergence; spatial epidemics; minimal position; times; LAW

摘要

We consider a branching random walk on Z started by n particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring 1 + theta/n. For t >= 0, we study M-nt, the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that M-nt/root n converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of M-nt. We also confirm that when theta > 0, the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky (Ann Probab 23(4):1748-1754, 1995). The rightmost position over all generations, M := sup(t) M-nt, is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Clumbel distribution when theta < 0.