Functional limit theorems for strongly subcritical branching processes in random environment
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 \(\mathbb{E}[m(Q_1)\log m(Q_1)]< \infty\) (``strong subcriticality'') and \(\mathbb{E}[Z_1\log^+ Z_1]<\infty\). 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.
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