THE PRECISE TAIL BEHAVIOR OF THE TOTAL PROGENY OF A KILLED BRANCHING RANDOM WALK

• 成果类型: Article
• 署名作者: Aidekon, Elie; Hu, Yueyun; Zindy, Olivier
• 署名单位: Eindhoven University of Technology; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; Universite Paris 13; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP842
• 发表日期: 2013
• 页码: 3786-3878
• 关键词: renewal theory; martingale convergence; brownian-motion; absorption; barrier; trees

摘要

Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/similar to aldous/Research/OP/brw.html].