Lecture Notes in Statistics

By: Wim Schoutens

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Paperback | 1 May 2000

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It has been known for a long time that there is a close connection between stochastic processes and orthogonal polynomials. For example, N. Wiener [112] and K. Ito [56] knew that Hermite polynomials play an important role in the integration theory with respect to Brownian motion. In the 1950s D. G. Kendall [66], W. Ledermann and G. E. H. Reuter [67] [74], and S. Kar lin and J. L. McGregor [59] established another important connection. They expressed the transition probabilities of a birth and death process by means of a spectral representation, the so-called Karlin-McGregor representation, in terms of orthogonal polynomials. In the following years these relation ships were developed further. Many birth and death models were related to specific orthogonal polynomials. H. Ogura [87], in 1972, and D. D. En gel [45], in 1982, found an integral relation between the Poisson process and the Charlier polynomials. Some people clearly felt the potential im portance of orthogonal polynomials in probability theory. For example, P. Diaconis and S. Zabell [29] related Stein equations for some well-known distributions, including Pearson's class, with the corresponding orthogonal polynomials. The most important orthogonal polynomials are brought together in the so-called Askey scheme of orthogonal polynomials. This scheme classifies the hypergeometric orthogonal polynomials that satisfy some type of differ ential or difference equation and stresses the limit relations between them.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Medicine Medicine in General Probability & Statistics Applied Mathematics Stochastics Reference, Information & Interdisciplinary Subjects Encyclopaedias & Reference Works Reference Works Algebra

Preface
The Askey Scheme of Orthogonal Polynomials	p. 1
Stochastic Processes	p. 15
Birth and Death Processes, Random Walks, and Orthogonal Polynomials	p. 31
Sheffer Systems	p. 45
Orthogonal Polynomials in Stochastic Integration Theory	p. 63
Stein Approximation and Orthogonal Polynomials	p. 91
Conclusion	p. 123
Distributions	p. 127
Tables of Classical Orthogonal Polynomials	p. 129
Table of Duality Relations Between Classical Orthogonal Polynomials	p. 139
Tables of Sheffer Systems	p. 141
Tables of Limit Relations Between Orthogonal Polynomials in the Askey Scheme	p. 145
References	p. 149
Index	p. 159
Table of Contents provided by Blackwell. All Rights Reserved.

Lecture Notes in Statistics

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