Path to survival for the critical branching processes in a random environment

Abstract

A critical branching process $\left\{Z_k,k=0,1,2,...\right\}$ in a random environment is considered. A conditional functional limit theorem for the properly scaled process $\left\{\log Z_{pu},0\le u<\infty \right\}$ is established under the assumptions $Z_n>0$ and $p\ll n$. It is shown that the limiting process is a Levy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.

18 pages