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›› 2011, Vol. 28 ›› Issue (3): 288-297.DOI: 10.7523/j.issn.2095-6134.2011.3.002

• Research Articles • Previous Articles     Next Articles

Some limit theorems on branching random walks in random environments

FANG Liang, HU Xiao-Yu   

  1. School of Mathematics, Graduate University, Chinese Academy of Sciences, Beijing 100049, China
  • Received:2010-04-12 Revised:2010-06-14 Online:2011-05-15

Abstract:

Suppose {Zn;n=0,1,2,…} is a branching random walk in the random environment, and ξ ={ξ0,ξ1,ξ2, …} is the environment process. Let Z(n,x) be the number of the nth generation located in the interval (-∞, x], fξn(s)=j=0 pξn(j)sj be the generating function of the distribution of the particle in the nth generation, and mξn=fξn' (1). We show that under the specific conditions, there exists a sequence of random variables {tn}, so that Z(n,tn)(∏i=0n-1mξi)-1 converges in L2. For branching random walks in varying environments, we have similar results.

Key words: branching process, branching random walks in random environments, branching random walks in varying environments

CLC Number: