Limit theorems for reduced branching processes in a random environment

Let $Z(n)$, $n=0,1\dots$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s)\dots$ and let $S_0=0$, $S_k=X_1+\dots+X_k$, $k\ge 1$, be the associated random walk with $X_i=\log f_{i-1}'(1)$, and let $\tau(n)$ be the leftmost point of minimum of $\{S_k\}_{k\ge 0}$ on the interval $[0,n]$. Denoting by $Z(k,n)$ the number of particles existing in the branching process at moment $k\le n$ and having nonempty offspring at time $n$ and assuming that the associated random walk satisfies the Spitzer–Doney condition $\mathbf{P}\{S_n>0\}\to\rho\in(0,1)$, $n\to\infty$, we show (under the quenched approach) that for each fixed $m=0,\pm 1,\pm 2,\dots$ the distribution of $Z(\tau(n)+m,n)$ given $Z(n)>0$ converges as $n\to\infty$ to a (random) discrete distribution which is proper with probability 1. On the other hand, if $m=m(n)\to\infty$ as $n\to\infty$, then to prove a conditional limit theorem for $Z(\tau(n)+m,n)$ given $Z(n)>0$, a scaling of $Z(\tau(n)+m,n)$ is needed by a function growing to infinity as $m\to\infty$.