Branching processes in random environment die slowly

Reference:

Vatutin, V. A. and Kyprianou, A. E., 2008. Branching processes in random environment die slowly. Discrete Mathematics & Theoretical Computer Science Proceedings, 2008, pp. 375-396.

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Official URL:

http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAI0125

Abstract

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.

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