The Annals of Probability

Weak Convergence to a Markov Chain with an Entrance Boundary: Ancestral Processes in Population Genetics

Peter Donnelly

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Abstract

We derive conditions under which a sequence of processes will converge to a (continuous-time) Markov chain with an entrance boundary. Our main application of this result is in proving weak convergence of the so-called population ancestral processes, associated with a wide class of exchangeable reproductive models, to a particular death process with an entrance boundary at infinity. This settles a conjecture of Kingman. We also prove weak convergence of the absorption times of many neutral genetics models to that of the Wright-Fisher diffusion, and convergence of population line-of-descent processes to another death process.

Article information

Source
Ann. Probab., Volume 19, Number 3 (1991), 1102-1117.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990336

Digital Object Identifier
doi:10.1214/aop/1176990336

Mathematical Reviews number (MathSciNet)
MR1112408

Zentralblatt MATH identifier
0732.92014

JSTOR
links.jstor.org

Subjects
Primary: 92A10
Secondary: 60F99: None of the above, but in this section 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
Entrance boundaries absorption times genetics models genealogical processes exchangeability ancestral numbers

Citation

Donnelly, Peter. Weak Convergence to a Markov Chain with an Entrance Boundary: Ancestral Processes in Population Genetics. Ann. Probab. 19 (1991), no. 3, 1102--1117. doi:10.1214/aop/1176990336. https://projecteuclid.org/euclid.aop/1176990336


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