Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T11:30:47.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean passage times for tridiagonal transition matrices and a two-parameter ehrenfest urn model

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Olaf Krafft  and
Martin Schaefer
Olaf Krafft
Affiliation:
Aachen University of Technology
Martin Schaefer*
Affiliation:
Aachen University of Technology
*
∗ Postal address for both authors: Institut für Statistik, RWTH Aachen, Wüllnerstr 3, 52056 Aachen, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

A two-parameter Ehrenfest urn model is derived according to the approach taken by Karlin and McGregor [7] where Krawtchouk polynomials are used. Furthermore, formulas for the mean passage times of finite homogeneous Markov chains with general tridiagonal transition matrices are given. In the special case of the Ehrenfest model they have quite a different structure as compared with those of Blom [2] or Kemperman [9].

Keywords

KRAWTCHOUK POLYNOMIALS

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Short Communications
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 964 - 970
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abramowitz, M. and Stegun, I. (1968) Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Blom, G. (1989) Mean transition times for the Ehrenfest urn model. Adv. Appl. Prob. 21, 479–480.Google Scholar
[3] Fritz, F.-J., Huppert, B. and Willems, W. (1979) Stochastische Matrizen. Springer-Verlag, Berlin.Google Scholar
[4] Kac, M. (1947) Random walk and the theory of Brownian motion. Amer. Math. Monthly 54, 369–391.CrossRefGoogle Scholar
[5] Karlin, S. (1968) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[6] Karlin, S. and Mcgregor, J. (1964) On some stochastic models in genetics. In Stochastic Models in Medicine and Biology, pp. 245–279, The University of Wisconsin Press, Madison.Google Scholar
[7] Karlin, S. and Mcgregor, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352–376.Google Scholar
[8] Kemeny, J. G. and Snell, J. L. (1969) Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
[9] Kemperman, J. H. B. (1961) The Passage Problem for a Stationary Markov Chain. The University of Chicago Press.CrossRefGoogle Scholar
[10] Szegö, G. (1978) Orthogonal Polynomials, 4th edn. Amer. Math. Soc. Colloquium Publications 23, Providence, RI.Google Scholar
[11] Van Beek, K. W. H. and Stam, A. J. (1987) A variant of the Ehrenfest model. Adv. Appl. Prob. 19, 995–996.Google Scholar