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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.

Details

Item Type Articles
CreatorsVatutin, V. A.and Kyprianou, A. E.
DepartmentsFaculty of Science > Mathematical Sciences
RefereedNo
StatusPublished
ID Code24219
Additional InformationProceedings paper from the Fifth Colloquium on Mathematics and Computer Science

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