Branching processes in random environment die slowly

1 Steklov Mathematical Institute 2 Department of Mathematical Sciences, University of Bath

Abstract : Let $Z n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f 0s,f 1s,\ldots,$ and let $S 0=0$, $S k=X 1+ \ldots +X k,k \geq 1$, be the associated random walk with $X i=\log f {i-1}^{\prime}1, \tau m,n$ be the left-most point of minimum of $\{S k,k \geq 0 \}$ on the interval $m,n$, and $T=\min \{ k:Z k=0\}$. Assuming that the associated random walk satisfies the Doney condition $PS n > 0 \to ho \in 0,1, n \to \infty$, we prove under the quenched approach conditional limit theorems, as $n \to \infty$, for the distribution of $Z {nt}, Z {\tau 0,nt}$, and $Z {\tau nt,n}, t \in 0,1$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau 0,n$ with respect to the point $nt$.

Keywords : critical branching process random environment limit theorems

Author: Vladimir Vatutin - Andreas Kyprianou -

Source: https://hal.archives-ouvertes.fr/