Stochastic Processes and Orthogonal Polynomials: 146 - Brossura

Schoutens, Wim

Brossura

ISBN 10: 038795015X ISBN 13: 9780387950150

Casa editrice: Springer, 2013

Riassunto

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.

Su questo libro

Contenuti

1 The Askey Scheme of Orthogonal Polynomials.- 2.1 Markov Processes.- 3 Birth and Death Processes, Random Walks, and Orthogonal Polynomials.- 4 Sheffer Systems.- 5 Orthogonal Polynomials in Stochastic Integration Theory.- Stein Approximation and Orthogonal Polynomials.- Conclusion.- A Distributions.- B Tables of Classical Orthogonal Polynomials.- B.1 Hermite Polynomials and the Normal Distribution.- B.2 Scaled Hermite Polynomials and the Standard Normal Distribution.- B.3 Hermite Polynomials with Parameter and the Normal Distribution.- B.4 Charlier Polynomials and the Poisson Distribution.- B.5 Laguerre Polynomials and the Gamma Distribution.- B.6 Meixner Polynomials and the Pascal Distribution.- B.7 Krawtchouk Polynomials and the Binomial Distribution.- B.8 Jacobi Polynomials and the Beta Kernel.- B.9 Hahn Polynomials and the Hypergeometric Distribution.- C Table of Duality Relations Between Classical Orthogonal Polynomials.- D Tables of Sheffer Systems.- D.1 Sheffer Polynomials and Their Generating Functions.- D.2 Sheffer Polynomials and Their Associated Distributions.- D.3 Martingale Relations with Sheffer Polynomials.- E Tables of Limit Relations Between Orthogonal Polynomials in the Askey Scheme.- References.

Product Description

Book by Schoutens Wim

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2013
• Lingua: Inglese
• ISBN 10: 038795015X
• ISBN 13: 9780387950150
• Rilegatura: Copertina flessibile
• Numero di pagine: 184
• Contatto del produttore: ProductSafety@springernature.com; ProductSafety@springernature.com; ; Poststr. 9; Darmstadt; 64293; Germania