Branching processes in random environment die slowly

Let Zn,n=0,1,..., be a branching process evolving in the random environment generated by a sequence of iid generating functions f0(s),f1(s),..., and let S0=0,Sk=X1+...+Xk,k≥1, be the associated random walk with Xi=logfi-1′(1), τ(m,n) be the left-most point of minimum of { Sk,k≥0} on the interval [m,n], and T=min { k:Zk=0} . Assuming that the associated random walk satisfies the Doney condition P( Sn>0) →ρ∈(0,1),n→∞, we prove (under the quenched approach) conditional limit theorems, as n→∞, for the distribution of Znt, Zτ(0,nt), and Zτ(nt,n), t∈(0,1), given T=n. It is shown that the form of the limit distributions essentially depends on the location of τ(0,n) with respect to the point nt.