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On a generalization of the ehrenfest urn model

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Holger Dette
Holger Dette*
Affiliation:
Technische Universität Dresden
*
∗Postal address: Institut für Mathematische Stochastik, Abteilung Mathematik, Technische Universität Dresden, Mommsenstr. 13, 01062 Dresden, Germany.
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Abstract

Krafft and Schaefer [14] considered a two-parameter Ehrenfest urn model and found the n-step transition probabilities using representations by Krawtchouk polynomials. For a special case of the model Palacios [17] calculated some of the expected first-passage times. This note investigates a generalization of the two-parameter Ehrenfest urn model where the transition probabilities pi,i+1 and pi,i+1 are allowed to be quadratic functions of the current state i. The approach used in this paper is based on the integral representations of Karlin and McGregor [9] and can also be used for Markov chains with an infinite state space.

Keywords

HAHN POLYNOMIALSEXPECTED FIRST-PASSAGE TIMESn-STEP TRANSITION PROBABILITIES

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994 , pp. 930 - 939
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported in part by the Deutsche Forschungsgemeinschaft.

References

[1] Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions , 9th edn. Dover, New York.Google Scholar
[2] Askey, R. (1975) Orthogonal Polynomials and Special Functions. SIAM, Philadelphia, PA.Google Scholar
[3] Bailey, N. T. (1984) The Elements of Stochastic Processes. Wiley, New York.Google Scholar
[4] Bingham, N. H. (1991) Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598–611.Google Scholar
[5] Chihara, T. S. (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
[6] Dette, H. (1990) A generalization of D- and D1-optimal designs in polynomial regression. Ann. Statist. 18, 1784–1804.Google Scholar
[7] Gasper, G. (1975) Positivity and special functions. In Theory and Applications of Special Functions , ed. Askey, R., pp. 375–433. Academic Press, New York.Google Scholar
[8] Ismail, M. E. H., Letessier, J. and Valent, G. (1989) Quadratic birth and death processes and associated continuous dual Hahn polynomials. SIAM J. Math. Anal. 20, 727–737.Google Scholar
[9] Karlin, S. and Mcgregor, J. (1959) Random walks. Illinois J. Math. 3, 66–81.Google Scholar
[10] Karlin, S. and Mcgregor, J. (1959) A characterization of birth and death processes. Proc. Nat. Acad. Sci. U.S.A. 45, 375–379.Google Scholar
[11] Karlin, S. and Mcgregor, J. (1961) The Hahn polynomials, formulas and an application. Scripta Math. 26, 33–46.Google Scholar
[12] Karlin, S. and Mcgregor, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352–376.CrossRefGoogle Scholar
[13] Kendall, D. G. (1958) Integral representations for Markov transition probabilities. Bull. Amer. Math. Soc. 64, 358–362.Google Scholar
[14] Krafft, O. and Schaefer, M. (1993) Mean passage times for triangular transition matrices and a two parameter Ehrenfest urn model. J. Appl. Prob. 30, 964–970.Google Scholar
[15] Letessier, J. and Valent, G. (1984) The generating function method for quadratic asymptotically symmetric birth and death processes. SIAM J. Appl. Math. 44, 773–783.Google Scholar
[16] Moran, P. A. P. (1958) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–72.Google Scholar
[17] Palacios, J. L. (1993) Fluctuation theory for the Ehrenfest urn via electronic networks. Adv. Appl. Prob. 25, 472–476.Google Scholar
[18] Rahman, M. (1978) A positive kernel for Hahn Eberlein polynomials. SIAM J. Math. 9, 891–905.Google Scholar
[19] Roehmer, B. and Valent, G. (1982) Solving the birth and death processes with quadratic asymptotically symmetric transition rates. SIAM J. Appl. Math. 42, 1020–1046.Google Scholar
[20] Skibinsky, M. (1968) Extreme nth moments on [0, 1] and the inverse of a moment space map. J. Appl. Prob. 5, 693–701.Google Scholar
[21] Skibinsky, M. (1986) Principal representations and canonical moment sequences for distributions on an interval. J. Math. Anal. Appl. 120, 95–120.Google Scholar
[22] Sloane, N. J. A. (1975) An introduction to association schemes and coding theory. In Theory and Application of Special Functions , ed. Askey, R., pp. 225–260. Academic Press, New York.Google Scholar
[23] Stanton, D. (1984) Orthogonal polynomials and Chevalley groups. In Special Functions: Group Theoretical Aspects and Applications , ed. Askey, R., Koornwinder, T. H. and Schempp, W., pp. 87–127. Reidel, Dordrecht.Google Scholar
[24] Szegö, G. (1975) Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ. 23, American Mathematical Society, Providence, RI.Google Scholar