Galton-Watson branching processes in a random environment I: Limit theorems

Let Z(n) be the number of individuals at time n in a branching process in a random environment generated by independent identically distributed random probability generating functions f(0)(s), f(1)(s),..., f(n)(s),.... Let X-i = log f'(i-1) (1), i = 0, 1, 2,...; S-0 = 0, S-n = X-1 + (...) X-n, n greater than or equal to 1. It is shown that if Z(n) is, in a sense, "critical," then there exists a limit in distribution lim(n-->infinity) exp { - min(0 (<=) under barj (<=) under barn) S-j} P{Z(n) > 0\f(0),..., f(n-1)} = zeta, where zeta is a proper random variable positive with probability 1. In addition, it is shown that for a "typical" realization of the environment the number of individuals Z(n) given {Z(n) > 0} grows as exp {S-n - min(0 (<=) under barj (<=) under barn)S(j)} (up to a positive finite random multiplier).