Branching Processes in Random Environment which Extinct at a Given Moment

#### C. Boeinghoff, E.E. Dyakonova, G. Kersting, V.A. Vatutin

2010, v.16, Issue 2, 329-350

ABSTRACT

Let $\{Z_{n}, n\geq 0\}$ be a critical branching process in random environment and let $T$ be its moment of extinction. Under the annealed approach we prove, as $n\rightarrow \infty$, a limit theorem for the number of particles in the process at moment $n$ given $T=n+1$ and a functional limit theorem for the properly scaled process $\{ Z_{nt}, \delta \leq t \leq 1-\delta \}$ given $T = n+1$ and $\delta \in (0,1/2)$.

Keywords: branching process,random environment,random walk,change ofmeasure,survival probability,functional limit theorem