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

A critical branching process $\{Z_k,k=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ in a random environment is considered. A conditional functional limit theorem for the properly scaled process $\{\log Z_{pu},0\le u<\mathrm{\infty }\}$ 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.