LIMIT THEOREMS FOR REDUCED BRANCHING PROCESSES IN A RANDOM ENVIRONMENT

Let Z(n), n = 0, 1,..., be a branching process evolving in the random environment generated by a sequence of independent identically distributed generating functions f(0)( s), f(1)( s),..., and let S-0 = 0, S-k = X-1+ ... + X-k, k >= 1, be the associated random walk with X-i = log f(i-1)(') (1), and let tau(n) be the leftmost point of minimum of {S-k} k >= 0 on the interval [ 0, n]. Denoting by Z(k, n) the number of particles existing in the branching process at moment k <= n and having nonempty offspring at time n and assuming that the associated random walk satisfies the Spitzer - Doney condition P{S-n > 0} -> rho epsilon (0, 1), n -> infinity, we show (under the quenched approach) that for each fixed m = 0, +/- 1, +/- 2,... the distribution of Z(tau(n) + m, n) given Z(n) > 0 converges as n ->infinity to a (random) discrete distribution which is proper with probability 1. On the other hand, if m = m(n) -> infinity as n -> infinity, then to prove a conditional limit theorem for Z(tau(n)+ m, n) given Z(n) > 0, a scaling of Z(t(n) + m, n) is needed by a function growing to infinity as m.8.