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 → ∞. We also prove the convergence of the conditional distribution of the process {n−12 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.