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Reduced Branching Processes with Very Heavy Tails

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Andreas N. Lagerås  and
Serik Sagitov
Andreas N. Lagerås*
Affiliation:
Chalmers University of Technology and Göteborg University
Serik Sagitov*
Affiliation:
Chalmers University of Technology
*
∗Postal address: Mathematical Sciences, Chalmers University of Technology, 412 96 Göteborg, Sweden.
∗∗Email address: [email protected]
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Abstract

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The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

Keywords

Reduced branching processcritical branching processheavy tailregular variation

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 190 - 200
Copyright
Copyright © Applied Probability Trust 2008 

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