THE SENETA-HEYDE SCALING FOR THE BRANCHING RANDOM WALK

• 成果类型: Article
• 署名作者: Aidekon, Elie; Shi, Zhan
• 署名单位: Sorbonne Universite; Universite Paris Cite; 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: 10.1214/12-AOP809
• 发表日期: 2014
• 页码: 959-993
• 关键词: galton-watson process; brownian-motion; martingale convergence; minimal position; traveling-waves; fixed-points; equation; trees; absorption; survival

摘要

We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609-631] in a one-dimensional supercritical branching random walk, and study the additive martingale (W-n). We prove that, upon the system's survival, n(1/2)W(n) converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou, of the derivative martingale.