Electronic Journal of Probability
- Electron. J. Probab.
- Volume 17 (2012), paper no. 63, 41 pp.
On Dirichlet eigenvectors for neutral two-dimensional Markov chains
We consider a general class of discrete, two-dimensional Markov chains modeling the dynamics of a population with two types, without mutation or immigration, and neutral in the sense that type has no influence on each individual's birth or death parameters. We prove that all the eigenvectors of the corresponding transition matrix or infinitesimal generator $\Pi$ can be expressed as the product of ``universal'' polynomials of two variables, depending on each type's size but not on the specific transitions of the dynamics, and functions depending only on the total population size. These eigenvectors appear to be Dirichlet eigenvectors for $\Pi$ on the complement of triangular subdomains, and as a consequence the corresponding eigenvalues are ordered in a specific way. As an application, we study the quasistationary behavior of finite, nearly neutral, two-dimensional Markov chains, absorbed in the sense that $0$ is an absorbing state for each component of the process.
Electron. J. Probab., Volume 17 (2012), paper no. 63, 41 pp.
Accepted: 18 August 2012
First available in Project Euclid: 4 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 39A14: Partial difference equations 47N30: Applications in probability theory and statistics 92D25: Population dynamics (general)
Hahn polynomials two-dimensional difference equation neutral Markov chain multitype population dynamics Dirichlet eigenvector Dirichlet eigenvalue quasi-stationary distribution Yaglom limit coexistence
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Champagnat, Nicolas; Diaconis, Persi; Miclo, Laurent. On Dirichlet eigenvectors for neutral two-dimensional Markov chains. Electron. J. Probab. 17 (2012), paper no. 63, 41 pp. doi:10.1214/EJP.v17-1830. https://projecteuclid.org/euclid.ejp/1465062385