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Zbl 1080.60079
Afanasyev, V.I.; Geiger, J.; Kersting, G.; Vatutin, V.A.
Functional limit theorems for strongly subcritical branching processes in random environment. (English)
[J] Stochastic Processes Appl. 115, No. 10, 1658-1676 (2005). ISSN 0304-4149

Let $Z_n$ denote the size of the $n$th 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 $Q_1,Q_2,Q_3,\dots$ are i.i.d. random distributions on $\{0,1,2,\dots\}$. For any distribution $q$ on $\{0,1,2,\dots\}$ define $m(q):= \sum_{y= 0}yq(\{y\})$. Suppose $\bbfE[m(Q_1)\log m(Q_1)]< \infty$ (``strong subcriticality'') and $\bbfE[Z_1\log^+ Z_1]<\infty$. 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 $(Z_n\mid Z_n> 0)$ converges in distribution to a positive recurrent Markov chain.
[Heinrich Hering (Rockenberg)]

MSC 2000:: *60J80 Branching processes
60G50 Sums of independent random variables
60F17 Functional limit theorems

Keywords: random walk; change of measure; positive recurrent Markov chain

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