Stochastic Processes and Orthogonal Polynomials
Springer Science & Business Media, 6 déc. 2012 - 184 pages
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.
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Table des matières
The Askey Scheme of Orthogonal Polynomials
Birth and Death Processes Random Walks and Orthogo
Orthogonal Polynomials in Stochastic Integration Theory
Stein Approximation and Orthogonal Polynomials
Table of Duality Relations Between Classical Orthogonal
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Expressions et termes fréquents
a f(a Askey scheme binomial process birth and death birth—death polynomials Brownian motion chaotic representation characteristic function Charlier polynomials classical orthogonal polynomials coefficient constant death process death rates defined denote differential equation diffusion discrete doubly limiting conditional dual Hahn polynomials duality relation Eacample Eſh(Z exponential functions f Gamma process Hahn polynomials Hermite polynomials hypergeometric type j|Xo Jacobi Karlin and McGregor Krawtchouk polynomials Laguerre polynomials Lévy measure Lévy process limiting conditional distribution limiting stationary distribution Markov process Meixner polynomials Meixner process Mn(a monic Notation orthogonal with respect parameters Poisson distribution Poisson process polynomial of degree Pr(X predictable representation Qn(a Racah polynomials random variable random walk representation property role Sheffer Polynomials ſº solution standard normal distribution Stein equation Stein operator Stein–Markov Stein's method stochastic integrals stochastic process Teugels martingales Theorem three-term recurrence relation transition probabilities