Stochastic Processes and Orthogonal Polynomials: 146 (Lecture Notes in Statistics) - Tapa blanda

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.

The book offers an accessible reference for researchers in the probability, statistics and special functions communities. It gives a variety of interdisciplinary relations between the two main ingredients of stochastic processes and orthogonal polynomials. It covers topics like time dependent and asymptotic analysis for birth-death processes and diffusions, martingale relations for Lévy processes, stochastic integrals and Stein's approximation method. Almost all well-known orthogonal polynomials, which are brought together in the so-called Askey Scheme, come into play. This volume clearly illustrates the powerful mathematical role of orthogonal polynomials in the analysis of stochastic processes and is made accessible for all mathematicians with a basic background in probability theory and mathematical analysis. Wim Schoutens is a Postdoctoral Researcher of the Fund for Scientific Research-Flanders (Belgium). He received his PhD in Science from the Catholic University of Leuven, Belgium.