PRODUCT PROCESSES IN VARYING ENVIRONMENTS

Abstract

A particular type of branching processes in varying environments is considered. It is assumed that all individuals of the same generation produce, given that the preceding generation is not extinct, randomly and independently of the past generations the same number of children. We show that the number of children in the nth generation normed by its expectation converges almost surely to a limit whose expectation is 0 or 1. We give a sufficient condition for convergence in quadratic mean to a limit whose mean is one. A nonclassical norming sequence of constants is defined so that the almost sure limit is finite greater than zero with probability 1. We also show, under certain circumstances, that the almost sure limit has infinite mean.