Reduced critical branching processes in random environment

Let Z(n), N = 0, 1, 2, ... be a critical branching process in random environment and Z(m, n), m <= n, the corresponding reduced process. We consider the case when the offspring generating functions are fractional linear and show that for any fixed m the conditional distribution of Z(m, n) given Z(n) > 0 converges to a non-trivial limit as n --> [infinity]. We also prove the convergence of the conditional distribution of the process {n-1/2 log Z([nt], n), 0 <= t <= 1} given Z(n) > 0 to the law of a transformation of the Brownian meander. Some applications of the above results to random walks in random environment are indicated.