On the concept of subcriticality and criticality and a ratio theorem for a branching process in a random environment

We consider a branching process (Zn) in a stationary and ergodic random environment ξ=(ξn). Athreya and Karlin (1971) proved the basic result about the concept of subcriticality and criticality, by showing that under the quenched law Pξ, the conditional distribution of Zn given the non-extinction at time n converges in law to a proper distribution on N+={1,2,⋯} in the subcritical case, and to the null distribution in the critical case, under the condition that the environment sequence is exchangeable. In this paper we first improve this basic result by removing the exchangeability condition on the environment, and by establishing a more general result about the conditional law of Zn given the non-extinction at time n+k for each fixed k≥0. As a by-product of the proof we also remove the exchangeability condition in another result of Athreya and Karlin (1971) for the subcritical case about the decay rate of the survival probability given the environment. We then establish a convergence theorem about the ratio Pξ(Zn=j)/Pξ(Zn=1), which can be applicable in each of the subcritical, critical, and supercritical cases.