Stochastic Processes and Orthogonal Polynomials

by Wim Schoutens

Overview

The book offers an accessible reference for researchers in the probabi lity, statistics and special functions communities. It gives a variety of interdisciplinary relations between the two main ingredients of st ochastic processes and orthogonal polynomials. It covers topics like t ime dependent and asymptotic analysis for birth-death processes and di vy processes, stochastic integral s and Stein's approximation method. Almost all well-known orthogonal p olynomials, which are brought together in the so-called Askey Scheme, come into play. This volume clearly illustrates the powerful mathemati cal role of orthogonal polynomials in the analysis of stochastic proce sses and is made accessible for all mathematicians with a basic backgr ound in probability theory and mathematical analysis.

Product Details

ISBN-13:	9780387950150
Publisher:	Springer New York
Publication date:	04/27/2000
Series:	Lecture Notes in Statistics , #146
Edition description:	2000
Pages:	184
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

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 Shastic 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.