We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q < 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q < 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

It is shown that a Galton-Watson process in varying environment, unlike a classical Galton-Watson process, can have an infinite (though at most countable) number of distinct rates of growth.