A Note on the Transience of Critical Branching Random Walks on the Line
A Note on the Transience of Critical Branching Random Walks on the Line
复制标题
关于线上临界分支随机游走的瞬态性的注解
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
M. Meiners
中科院分区：
文献类型：
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作者：
G. Alsmeyer;M. Meiners
Gantert和Muller（2006）通过在分支马尔可夫链的更一般框架内分析这个问题并利用李雅普诺夫函数证明了整数格上的临界分支随机游动（BRW）是瞬态的。本文的主要目的是说明如何在加权分支过程理论中很好地导出同样的结果，甚至推广到非格点情形。这是通过分析某些相关的随机加权位置的措施，在采取预期，提供了一个有用的连接到建立良好的理论，普通的随机游动与独立同分布。增量。最后一节利用Hu和Shi（2008）最近的工作，简要讨论了临界BRW中最左边和最右边粒子随着时间趋于无穷大的渐近行为。
Gantert and Muller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).