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Characterization by orthogonal polynomial systems of finite Markov chains

Part of: History of mathematics and mathematicians Markov processes

Published online by Cambridge University Press:  14 July 2016

E. Seneta
E. Seneta*
Affiliation:
University of Sydney
*
1Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email: [email protected]
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Abstract

The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

Keywords

Orthogonal polynomialsreversibilityMarkov chainsspectrumeigenvectorsChebyshevHahnKrawtchoukRacahMoran mutation modelhypergeometricself-dualityBernoulli-LaplaceEhrenfestVere-Jonesbirth-and-death processes

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 01A60: 20th century
Type
Markov chains
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 42 - 52
Copyright
Copyright © Applied Probability Trust 2001 

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