PATH TO SURVIVAL FOR THE CRITICAL BRANCHING PROCESSES IN A RANDOM ENVIRONMENT

摘要

A critical branching process {Z(k), k = 0, 1, 2,...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Z(pu), 0 <= u < infinity} is established under the assumptions that Z(n) > 0 and p 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.