Journal of Applied Probability

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

Martin Möhle and Morihiro Notohara

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We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius ress(P|𝓁²(𝜋)) of P|𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−NNlim supi→+∞(P(i,i+m)P*(i+m,i)1∕2<1. Moreover, ress(P|𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

Article information

J. Appl. Probab., Volume 53, Number 3 (2016), 953-956.

First available in Project Euclid: 13 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)

Convergence Markov chain population genetics separation of time-scales


Möhle, Martin; Notohara, Morihiro. An extension of a convergence theorem for Markov chains arising in population genetics. J. Appl. Probab. 53 (2016), no. 3, 953--956.

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