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First return probabilities of birth and death chains and associated orthogonal polynomials

Author: Holger Dette
Journal: Proc. Amer. Math. Soc. 129 (2001), 1805-1815
MSC (1991): Primary 60J15; Secondary 33C45
Published electronically: November 2, 2000
MathSciNet review: 1814114
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Abstract: For a birth and death chain on the nonnegative integers, integral representations for first return probabilities are derived. While the integral representations for ordinary transition probabilities given by Karlin and McGregor (1959) involve a system of random walk polynomials and the corresponding measure of orthogonality, the formulas for the first return probabilities are based on the corresponding systems of associated orthogonal polynomials. Moreover, while the moments of the measure corresponding to the random walk polynomials give the ordinary return probabilities to the origin, the moments of the measure corresponding to the associated polynomials give the first return probabilities to the origin.

As a by-product we obtain a new characterization in terms of canonical moments for the measure of orthogonality corresponding to the first associated orthogonal polynomials. The results are illustrated by several examples.

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Additional Information

Holger Dette
Affiliation: Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany

Keywords: Birth and death chain, spectral measure, orthogonal polynomials, associated polynomials, canonical moments
Received by editor(s): April 8, 1999
Received by editor(s) in revised form: September 7, 1999
Published electronically: November 2, 2000
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2000 American Mathematical Society

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