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Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Jyy-I Hong
Jyy-I Hong*
Affiliation:
Waldorf College
*
∗ Postal address: Department of Mathematics, Waldorf College, 106 South 6th Street, Forest City, Iowa 50436, USA. Email address: hongjyyi@gmail.com
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Abstract

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We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

Keywords

Branching processcoalescencesubcriticalBellmanHarrisage dependentline of descent

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 576 - 591
Copyright
© Applied Probability Trust 

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