  We consider a subcritical branching process in an i.i.d. random environment,
in which one immigrant arrives at each generation. We consider the event $%
\mathcal{A}_{i}(n)$ that all individuals alive at time $n$ are offspring of the
immigrant which joined the population at time $i$ and investigate the
asymptotic probability of this extreme event when $n\to\infty$ and $i$ is
either fixed, or the difference $n-i$ is fixed, or $\min(i,n-i)\to\infty.$ To
deduce the desired asymptotics we establish some limit theorems for random
walks conditioned to be nonnegative or negative.
