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Functional limit theorems for strongly subcritical branching processes in random environment. (English) Zbl 1080.60079
Let Z n denote the size of the nth generation of a one-type, discrete-time branching process in an i.i.d. environment, i.e., Z n is the sum of Z n-1 random variables, each distributed according to Q n , where Q 1 ,Q 2 ,Q 3 , are i.i.d. random distributions on {0,1,2,}. For any distribution q on {0,1,2,} define m(q):= y=0 yq({y}). Suppose 𝔼[m(Q 1 )logm(Q 1 )]< (“strong subcriticality”) and 𝔼[Z 1 log + Z 1 ]<. Then, conditioned on survival in a sufficiently distant future, the environments remain in the limit independent, in contrast to the situation in case of weaker forms of subcriticality. Furthermore, the conditioned process (Z n Z n >0) converges in distribution to a positive recurrent Markov chain.
MSC:
60J80Branching processes
60G50Sums of independent random variables; random walks
60F17Functional limit theorems; invariance principles