Branching processes in random environment die slowly


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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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.


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


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