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On Dirichlet eigenvectors for neutral two-dimensional Markov chains


 
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1. Title Title of document On Dirichlet eigenvectors for neutral two-dimensional Markov chains
 
2. Creator Author's name, affiliation, country Nicolas Champagnat; Nancy Université and INRIA Nancy - Grand Est; France
 
2. Creator Author's name, affiliation, country Persi Diaconis; Stanford University; United States
 
2. Creator Author's name, affiliation, country Laurent Miclo; Université Paul Sabatier, Toulouse; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Hahn polynomials; two-dimensional difference equation; neutral Markov chain; multitype population dynamics; Dirichlet eigenvector; Dirichlet eigenvalue; quasi-stationary distribution; Yaglom limit; coexistence
 
3. Subject Subject classification Primary: 60J10, 60J27; secondary: 15A18, 39A14, 47N30, 92D25.
 
4. Description Abstract 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) ANR (French national research agency)
 
7. Date (YYYY-MM-DD) 2012-08-18
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/1830
 
10. Identifier Digital Object Identifier 10.1214/EJP.v17-1830
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 17
 
12. Language English=en en
 
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