Reduced processes evolving in a mixed environment

Abstract

Abstract We consider a two-type decomposable branching process where type 1 particles may produce particles of types 1 and 2 while type 2 particles can give birth only to type 2 particles. Let i = 1, 2 be the number of type i particles existing in the process at moment m < n and having a positive number of descendants at moment n. Assuming that particles of the first type evolve in a random environment and particles of the second type evolve in a constant environment we investigate the distribution of the random vector when and