Stochastic Processes and Orthogonal Polynomials Paperback / softback
by Wim Schoutens
Part of the Lecture Notes in Statistics series
Paperback / softback
Description
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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Item not Available
- Format:Paperback / softback
- Pages:184 pages, XIII, 184 p.
- Publisher:Springer-Verlag New York Inc.
- Publication Date:27/04/2000
- Category:
- ISBN:9780387950150
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Information
-
Item not Available
- Format:Paperback / softback
- Pages:184 pages, XIII, 184 p.
- Publisher:Springer-Verlag New York Inc.
- Publication Date:27/04/2000
- Category:
- ISBN:9780387950150