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

Journal of Applied Probability

Volumn 33, Issue 4, 1996, Pages 1033-1052

  Dette, Holger ^a  

^a RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

Ehrenfest urn; Generating function; Orthogonal polynomials; Random walk; Spectral measure; Stieltjes transform

Indexed keywords

EID: 0030359682     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0021900200100464     Document Type: Article

References (23)

1. ABRAMOWITZ, M. AND STEGUN, I. A. (1964) Handbook of Mathematical Functions. Dover, New York.
2. 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).
3. BINGHAM, N. H. (1991) Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598-611.
4. CHANG, F. C., KEMPERMAN, J. H. B. AND STUDDEN, W. J. (1993) A normal limit theorem for moment sequences. Ann. Prob. 21, 1295-1305.
5. CHIHARA, T. S. (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.
6. DETTE, H. (1994) On a generalization of the Ehrenfest urn model. J. Appl. Prob. 31, 930-939.
7. FELLER, W. (1967) An Introduction to Probability Theory and its Applications. Vol. I. 3rd edn. Wiley, New York.
8. FREUD, G. (1971) Orthogonal Polynomials. Pergamon, Oxford.
9. HARRIS, T. E. (1952) First passage and recurrence distributions. Trans. Amer. Math. Soc. 73, 471-486.
10. 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.
11. KARLIN, S. AND MCGREGOR, J. (1959) Random walks. Illinois J. Math. 3, 66-81.
12. 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.
13. KARLIN, S. AND MCGREGOR, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352-376.
14. KARLIN, S. AND TAYLOR, H. M. (1975) A First Course in Stochastic Processes. 2nd edn. Academic Press, New York.
15. KENDALL, D. G. (1958) Integral representations for Markov transition probabilities. Bull. Amer. Math. Soc. 64, 358-362.
16. 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.
17. KERSTING, G. (1974) Strong ratio limit property and R-recurrence of reversible Markov chains. Prob. Theory Rel Fields 30, 343-356.
18. 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.
19. LEGUESDRON, P., PELLAUMAIL, J., RUBINO, G. AND SERICOLA, B. (1993) Transient analysis of the M/M/1 queue. Adv. Appl. Prob. 25, 702-713.
20. PERCUS, O. (1985) Phase transition in one-dimensional random walk with partially reflecting boundaries. Adv. Appl. Prob. 17, 594-606.
21. RIVLIN, T. J. (1990) Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory. 2nd edn. Wiley-Interscience, New York.
22. SZEGÖ, G. (1975) Orthogonal Polynomials (Amer. Math. Soc. Colloq. 23). American Mathematical Society, Providence, R1.
23. WALL, H. S. (1948) Analytic Theory of Continued Fractions. Van Nostrand, New York.