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Quasi-stationary distributions of a pair of Markov chains related to time evolution of a DNA locus

Part of: Miscellaneous applications of operator theory

Published online by Cambridge University Press:  01 July 2016

A. Bobrowski
A. Bobrowski*
Affiliation:
Lublin University of Technology
*
∗ Postal address: Department of Mathematics, Faculty of Electrical Engineering, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland. Email address: adambob@antenor.pol.lublin.pl
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Abstract

We consider a pair of Markov chains representing statistics of the Fisher-Wright-Moran model with mutations and drift. The chains have absorbing state at 0 and are related by the fact that some random time τ ago they were identical, evolving as a single Markov chain with values in {0,1,…}; from that time on they began to evolve independently, conditional on a state at the time of split, according to the same transition probabilities. The distribution of τ is a function of deterministic effective population size 2N(·). We study the impact of demographic history on the shape of the quasi-stationary distribution, conditional on nonabsorption at the margin (where one of the chains is at 0), and on the speed with which the probability mass escapes to the margin.

Keywords

Fisher-Wright-Moran modelquasi-stationary distributionMarkov chainmutationgenetic driftcoalescencemicrosatellite

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 47N60: Applications in chemistry and life sciences
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 57 - 77
Copyright
Copyright © Applied Probability Trust 2004 

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