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Duality and Asymptotics for a Class of Nonneutral Discrete Moran Models

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Thierry Huillet  and
Martin Möhle
Thierry Huillet*
Affiliation:
Université de Cergy-Pontoise
Martin Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
∗Postal address: Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8098 et Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France. Email address: thierry.huillet@u-cergy.fr
∗∗Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: martin.moehle@uni-tuebingen.de
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Abstract

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A Markov chain X with finite state space {0,…,N} and tridiagonal transition matrix is considered, where transitions from i to i-1 occur with probability (i/N)(1-p(i/N)) and transitions from i to i+1 occur with probability (1-i/N)p(i/N). Here p:[0,1]→[0,1] is a given function. It is shown that if p is continuous with p(x)≤p(1) for all x∈[0,1] then, for each N, a dual process Y to X (with respect to a specific duality function) exists if and only if 1-p is completely monotone with p(0)=0. A probabilistic interpretation of Y in terms of an ancestral process of a mixed multitype Moran model with a random number of types is presented. It is shown that, under weak conditions on p, the process Y, properly time and space scaled, converges to an Ornstein–Uhlenbeck process as N tends to ∞. The asymptotics of the stationary distribution of Y is studied as N tends to ∞. Examples are presented involving selection mechanisms. results.

Keywords

Ancestral processcomplete monotonicitydescendantsdualitymultitype Moran modelOrnstein–Uhlenbeck process

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D10: Genetics 92D25: Population dynamics (general)
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 866 - 893
Copyright
Copyright © Applied Probability Trust 2009 

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