Reduced critical branching processes in random environment

Abstract

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-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. © 1997 Elsevier Science B.V.