Electronic Communications in Probability

A spectral decomposition for a simple mutation model

Martin Möhle

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Abstract

We consider a population of $N$ individuals. Each individual has a type belonging to some at most countable type space $K$. At each time step each individual of type $k\in K$ mutates to type $l\in K$ independently of the other individuals with probability $m_{k,l}$. It is shown that the associated empirical measure process is Markovian. For the two-type case $K=\{0,1\}$ we derive an explicit spectral decomposition for the transition matrix $P$ of the Markov chain $Y=(Y_n)_{n\ge 0}$, where $Y_n$ denotes the number of individuals of type $1$ at time $n$. The result in particular shows that $P$ has eigenvalues $(1-m_{0,1}-m_{1,0})^i$, $i\in \{0,\ldots ,N\}$. Applications to mean first passage times are provided.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 15, 14 pp.

Dates
Received: 11 August 2018
Accepted: 26 February 2019
First available in Project Euclid: 21 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1553133704

Digital Object Identifier
doi:10.1214/19-ECP222

Mathematical Reviews number (MathSciNet)
MR3933039

Zentralblatt MATH identifier
1412.60106

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
eigenvalues eigenvectors empirical measure process finite Markov chain first passage time mixing time mutation model potential theory product chain random walk on the hypercube spectral analysis

Rights
Creative Commons Attribution 4.0 International License.

Citation

Möhle, Martin. A spectral decomposition for a simple mutation model. Electron. Commun. Probab. 24 (2019), paper no. 15, 14 pp. doi:10.1214/19-ECP222. https://projecteuclid.org/euclid.ecp/1553133704


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