随机环境中的分枝随机游动的若干极限定理

Some limit theorems on branching random walks in random environments

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=0pξn(j)sj为第n代个体的生成函数,mξn=f′ξn(1).证明了在特定条件下,存在随机序列{tn}使得Z(n,tn)(∏n-1i=0mξi)-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。

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 = f ξ＇n（1）.We show that under the specific conditions,there exists a sequence of random variables {tn},so that Z（n,tn）（∏ n-1 i = 0 m ξi）-1 converges in L2.For branching random walks in varying environments,we have similar results.