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Characterization by orthogonal polynomial systems of finite Markov chains

  • E. Seneta (a1)
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 a 0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij } satisfying p 00 + p 01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

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Corresponding author
1 Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email: e.seneta@maths.usyd.edu.au
References
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Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Cambridge University Press.
Askey, R. and Wilson, J. (1979). A set of polynomials that generalize the Racah coefficients on 6 - j symbols. SIAM J. Math. Anal. 10, 10081016.
Chebyshev, P. L. (1875). Sur l'interpolation des valeurs équidistants. In Œuvres de P. L. Tchebychef , Tome II, eds Markoff, A. and Sonin, N., Chelsea, New York, 219242. (1962 reprint. Volumes I and II originally published in St. Petersburg, 1899-1907).
Donnelly, P., Lloyd, P. and Sudbury, A. (1994). Approach to stationarity of the Bernoulli-Laplace diffusion model. Adv. Appl. Probab. 26, 715727.
Dunkl, C. F. and Ramirez, D. E. (1974). Krawtchouk polynomials and the symmetrization of hypergroups. SIAM J. Math. Anal. 5, 351366.
Eagleson, G. K. (1969). A characterization theorem for positive definite sequences on the Krawtchouk polynomials. Austral. J. Statist , ll, 2938.
Hahn, W. (1949). Uber Orthogonalpolynome, die q-differenzengleichungen genügen. Math. Nachr. 2, 434.
Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967). A refinement of two theorems in the theory of branching processes. Teor. Veroyatnost. i Primenen. 12, 297301.
Hoare, M. R. and Rahman, M. (1983). Cumulative Bernoulli trials and Krawtchouk processes. Stoch. Proc. Applic. 16, 113139.
Karlin, S. and Mcgregor, J. L. (1961). The Hahn polynomials, formulas and an application. Scripta Math. 26, 3346.
Karlin, S. and Mcgregor, J. L. (1962). On a genetics model of Moran. Proc. Cambridge Philos. Soc. 58, 299–231.
Karlin, S. and Mcgregor, J. L. (1965). Ehrenfest urn models. J. Appl. Probab. 2, 352376.
Kemeny, J. G. and Snell, L. J. (1960). Finite Markov Chains. Van Nostrand, Princeton, NJ.
Kendall, M. G. and Stuart, A. (1963). The Advanced Theory of Statistics, Vol. 1, Distribution Theory. Griffin, London.
Moran, P. A. P. (1960). The Statistical Processes of Evolutionary Theory. Oxford University Press.
Seneta, E. (1997). M. Krawtchouk (1892-1942), Professor of Mathematical Statistics. Theory Stoch. Proc. 3, 388392.
Seneta, E. (1998). Characterization of Markov chains by orthogonal polynomial systems. In Conference Materials, 7th Internat. Sci. Kravchuk Conf., 14-16 May 1998, Kyiv , Vipol, Kyiv, 454457.
Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3, 403434.
Sneddon, I. N. (1961). Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd, Edinburgh.
Vere-Jones, D. (1962). Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford (2) 13, 728.
Vere-Jones, D. (1967a). Ergodic properties of non-negative matrices, I. Pacific J. Math. 22, 361386.
Vere-Jones, D. (1967b). The infinite divisibility of a bivariate gamma distribution. Sankhya A 29, 421422.
Vere-Jones, D. (1968). Ergodic properties of non-negative matrices, II. Pacific J. Math. , 26, 601620.
Vere-Jones, D. (1971). Finite bivariate distributions and semigroups of non-negative matrices. Quart. J. Math. Oxford (2) , 22, 247270.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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