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On the generating functions of a random walk on the non-negative integers

Published online by Cambridge University Press:  14 July 2016

Holger Dette
Holger Dette*
Affiliation:
Technische Universität Dresden
*
∗ Postal address: Fakultät und Institut für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany.
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Abstract

In the random walk whose state space is a subset of the non-negative integers explicit representations for the generating functions of the n-step transition and the first return probabilities are obtained. These representations involve the Stieltjes transform of the spectral measure of the process and the corresponding orthogonal polynomials. Several examples are given in order to illustrate the application of the results.

Keywords

RANDOM WALKGENERATING FUNCTIONSPECTRAL MEASURESTIELTJES TRANSFORMORTHOGONAL POLYNOMIALSEHRENFEST URN

MSC classification

Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 1033 - 1052
Copyright
Copyright © Applied Probability Trust 1996 

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References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Askey, R. and Ismail, M. E. H. (1984) Recurrence relations, continued fractions and orthogonal polynomials. Mem. Amer. Math. Soc. 48, No. 300 (second of 5 numbers).Google Scholar
Bingham, N. H. (1991) Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598–611.Google Scholar
Chang, F. C., Kemperman, J. H. B. and Studden, W. J. (1993) A normal limit theorem for moment sequences. Ann. Prob. 21, 1295–1305.CrossRefGoogle Scholar
Chihara, T. S. (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
Dette, H. (1994) On a generalization of the Ehrenfest urn model. J. Appl. Prob. 31, 930–939.Google Scholar
Feller, W. (1967) An Introduction to Probability Theory and its Applications. Vol. I. 3rd edn. Wiley, New York.Google Scholar
Freud, G. (1971) Orthogonal Polynomials. Pergamon, Oxford.Google Scholar
Harris, T. E. (1952) First passage and recurrence distributions. Trans. Amer. Math. Soc. 73, 471–486.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1957) The differential equations of birth-and-death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1959) Random walks. Illinois J. Math. 3, 66–81.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1964) On some stochastic models in genetics. In Stochastic Models in Medicine and Biology. University of Wisconsin Press. pp. 245–279.Google Scholar
Karlin, S. and McGregor, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352–376.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. 2nd edn. Academic Press, New York.Google Scholar
Kendall, D. G. (1958) Integral representations for Markov transition probabilities. Bull. Amer. Math. Soc. 64, 358–362.CrossRefGoogle Scholar
Kent, J. T. and Longford, N. T. (1983) An eigenvalue decomposition for first hitting times in random walks. Prob. Theory Rel. Fields 63, 71–84.Google Scholar
Kersting, G. (1974) Strong ratio limit property and R-recurrence of reversible Markov chains. Prob. Theory Rel Fields 30, 343–356.Google Scholar
Krafft, O. and Schaefer, M. (1993) Mean passage times for tridiagonal transition matrices and a two parameter Ehrenfest urn model. J. Appl. Prob. 30, 964–970.CrossRefGoogle Scholar
Leguesdron, P., Pellaumail, J., Rubino, G. and Sericola, B. (1993) Transient analysis of the M/M/1 queue. Adv. Appl. Prob. 25, 702–713.Google Scholar
Percus, O. (1985) Phase transition in one-dimensional random walk with partially reflecting boundaries. Adv. Appl. Prob. 17, 594–606.CrossRefGoogle Scholar
Rivlin, T. J. (1990) Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory. 2nd edn. Wiley-Interscience, New York.Google Scholar
Szegö, G. (1975) Orthogonal Polynomials (Amer. Math. Soc. Colloq. 23). American Mathematical Society, Providence, RI.Google Scholar
Wall, H. S. (1948) Analytic Theory of Continued Fractions. Van Nostrand, New York.Google Scholar