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An extension of a convergence theorem for Markov chains arising in population genetics

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  24 October 2016

Martin Möhle  and
Morihiro Notohara
Martin Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
Morihiro Notohara*
Affiliation:
Nagoya City University
*
* Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: martin.moehle@uni-tuebingen.de
** Postal address: Graduate School of Natural Sciences, Nagoya City University, Mizuho, Nagoya, 467-8501, Japan. Email address: noto@nsc.nagoya-cu.ac.jp
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Abstract

An extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (XN(r))r be a Markov chain with the same finite state space S and transition matrix ΠN=I+dNBN, where I is the unit matrix, Q a generator matrix, (BN)N a sequence of matrices, limN℩∞cN= limN→∞dN=0 and limN→∞cN∕dN=0. Suppose that the limits P≔limm→∞(I+dNQ)m and G≔limN→∞PBNP exist. If the sequence of initial distributions PXN(0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (XN([t∕cN))t≥0 converge to those of the Markov process (Xt)t≥0 with initial distribution μ, transition matrix PetG and limN→∞(I+dNQ+cNBN)[t∕cN]

Keywords

ConvergenceMarkov chainpopulation geneticsseparation of time-scales

MSC classification

Primary: 60F05: Central limit and other weak theorems 92D10: Genetics
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 953 - 956
Copyright
Copyright © Applied Probability Trust 2016 

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