Total progeny in killed branching random walk

• 成果类型: Article
• 署名作者: Addario-Berry, L.; Broutin, N.
• 署名单位: McGill University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0299-2
• 发表日期: 2011
• 页码: 265-295
• 关键词: brownian-motion; optimal path; trees; probabilities; deviations

摘要

We consider a branching random walk for which the maximum position of a particle in the n'th generation, R-n, has zero speed on the linear scale: R-n/n -> 0 as n -> infinity. We further remove (kill) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite (Gantert and Muller in Markov Process. Relat. Fields 12:805-814, 2006; Hu and Shi in Ann. Probab. 37(2):742-789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388-412, 1998; and Power laws and killed branching random walks) that E[Z] < infinity while E[Z log Z] = infinity. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.