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\leq 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.