Limit theorems for functionals of a branching process in random environment starting with a large number of particles

Assume given a sequence Z^ (k) =Z₈^ (k), i=0, 1, …\}, k=1, 2, …, of critical branching processes in random environment which are only different from one another by the size k of the initial generation. Assume that the step of the associated random walk is in the zone of attraction of a stable law. In the case when k=k (n), where n is a positive integer parameter and k (n) grows with n in a certain special way, limit theorems as n are established for a process with continuous time constructed from the Z^ (k (n) ) and for the logarithm of this process. In addition, limit theorems are proved for the extinction time of the process Z^ (k (n) ), the maximum of this process, and the total number of particles. Bibliography: 10 titles.