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

Branching processes in random environment (BPRE's) describe development of such populations where the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Conditioned on the environment individuals reproduce independently of each other. It happens that the asymptotic properties of such processes are closely related to the properties of some random walks constructed by the characteristics of the initial BPRE. This connection is used in a number of papers written by the speaker and his colleagues and allows to develop a unique approach in studying BPRE's.
We give in this talk a detailed description of the connection and show how one can use such a connection to describe the properties of the initial part of the trajectory of a critical BPRE given its survival for a time essentially exceeding the length of the part under consideration.