Stochastic Processes and Orthogonal Polynomials (Lecture Notes in Statistics)

By Wim Schoutens

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.

• Format: Paperback
• Language: English
• ISBN: 038795015X
• ISBN13: 9780387950150
• Release Date: April 2000
• Publisher: Springer
• Length: 184 Pages
• Weight: 0.59 lbs.
• Dimensions: 0.4" x 6.1" x 9.2"