收稿日期: 2010-04-12
修回日期: 2010-06-14
网络出版日期: 2011-05-15
基金资助
国家自然科学基金(10871200)资助
Some limit theorems on branching random walks in random environments
Received date: 2010-04-12
Revised date: 2010-06-14
Online published: 2011-05-15
假设{Zn;n=0,1,2,…}是一个随机环境中的分枝随机游动(即质点在产生后代的过程中,还作直线上随机游动), ξ ={ξ0,ξ1,ξ2,…} 为环境过程. 记Z(n,x)为落在区间(-∞, x]中的第n代质点的个数,fξn(s)=∑j=0∞ pξn(j)sj 为第n代个体的生成函数, mξn=fξn' (1). 证明了在特定条件下,存在随机序列{tn}使得Z(n,tn)(∏i=0n-1mξi)-1均方收敛到一个随机变量.对于依赖于代的分枝随机游动,仍有类似的结论.
关键词： 分枝过程; 随机环境中的分枝随机游动; 依赖于代的分枝随机游动
方亮 , 胡晓予 . 随机环境中的分枝随机游动的若干极限定理[J]. 中国科学院大学学报, 2011 , 28(3) : 288 -297 . DOI: 10.7523/j.issn.2095-6134.2011.3.002
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.
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