On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements

• 成果类型: Article
• 署名作者: Chen, Xinxin; He, Hui
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Beijing Normal University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0891-4
• 发表日期: 2019
• 页码: 255-307
• 关键词: exact convergence-rates; random environment; limit-theorem; particles; sums

摘要

Given a branching random walk on R started from the origin, where the tail of the branching law decays at least exponentially fast and the offspring number is at least one, let Z(n)(.) be the counting measure which counts the number of individuals at the n-th generation located in a given set. Under some mild conditions, it is known (Biggins in Stoch. Process. Appl. 34:255-274, 1990) that for any interval A subset of R, Z(n)(root nA)/Z(n)(R) converges a.s. to nu(A), where nu is the standard Gaussian measure. In this work, we investigate the convergence rates of P(Z(n)(root nA)/Z(n)(R) - nu(A) > Delta), for Delta is an element of (0, 1 - nu(A)). We consider both the Schroder case, where the offspring number could be one, and the Bottcher case, where the offspring number is at least two.