Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T02:43:52.905Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Limit theorems

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
Get access Rights & Permissions [Opens in a new window]

Abstract

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

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ethier, S. N. and Kurtz, T. G. (1986).Markov Processes: Characterization and Convergence.John Wiley,New York.CrossRefGoogle Scholar
[2] Hössjer, O. (2011).Coalescence theory for a general class of structured populations with fast migration.Adv. Appl. Prob. 43,1027–1047.Google Scholar
[3] Kaj, I.,Krone, S. M. and Lascoux, M. (2001).Coalescent theory for seed bank models.J. Appl. Prob. 38,285–300.Google Scholar
[4] Möhle, M. (1998).A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing.Adv. Appl. Prob. 30,493–512.Google Scholar
[5] Möhle, M. (1998).Coalescent results for two-sex population models.Adv. Appl. Prob. 30,513–520.Google Scholar
[6] Nordborg, M. and Krone, S. M. (2001).Separation of time scales and convergence to the coalescent in structured populations. In Modern Developments in Theoretical Population Genetics, eds M.\ Slatkin and M. Veuille,Oxford University Press, pp. 194–232.Google Scholar
[7] Pollak, E. (2011).Coalescent theory for age-structured random mating populations with two sexes.Math. Biosci. 233,126–134.Google Scholar
[8] Sampson, K. Y. (2006).Structured coalescent with nonconservative migration.J. Appl. Prob. 43,351–362.Google Scholar