Waves in Reduced Branching Processes in a Random Environment

Let $Z(n)$, $n=0,1…,$ be a branching process evolving in the random environment generated by a sequence of independent identically distributed generating functions $f_{0}(s),f_{1}(s),…,$ and let $S_{0}=0$, $S_{k}=X_{1}+…+X_{k}$, $k\ge1,$ be the associated random walk with $X_{i}=\log f_{i-1}'(1),$ and $\tau (n)$ be the leftmost point of the minimum of $\{ S_{k}$,$k\ge0\} $ on the interval $[0,n]$. Denoting by $Z(k,m)$ the number of particles existing in the branching process at the time moment $k\le m$ which have nonempty offspring at the time moment $m$, and assuming that the associated random walk satisfies the Doney condition $P(S_{n}>0)\to \rho \in (0,1)$, $n\to\infty$, we prove (under the quenched approach) conditional limit theorems, as $n\to\infty$, for the distribution of $Z(nt_{1},nt_{2})$, $0<t_{1}<t_{2}<1,$ given $Z(n)>0$. It is shown that the form of the limit distributions essentially depends on the position of $\tau (n)$ with respect to the interval $[nt_{1},nt_{2}].$