Stochastic Processes And Orthogonal Polynomials
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Author | : Wim Schoutens |
Publisher | : Springer Science & Business Media |
Total Pages | : 170 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461211700 |
Download Stochastic Processes and Orthogonal Polynomials Book in PDF, Epub and Kindle
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.
Author | : Manuel Domínguez de la Iglesia |
Publisher | : Cambridge University Press |
Total Pages | : 348 |
Release | : 2021-10-21 |
Genre | : Mathematics |
ISBN | : 1009035207 |
Download Orthogonal Polynomials in the Spectral Analysis of Markov Processes Book in PDF, Epub and Kindle
In pioneering work in the 1950s, S. Karlin and J. McGregor showed that probabilistic aspects of certain Markov processes can be studied by analyzing orthogonal eigenfunctions of associated operators. In the decades since, many authors have extended and deepened this surprising connection between orthogonal polynomials and stochastic processes. This book gives a comprehensive analysis of the spectral representation of the most important one-dimensional Markov processes, namely discrete-time birth-death chains, birth-death processes and diffusion processes. It brings together the main results from the extensive literature on the topic with detailed examples and applications. Also featuring an introduction to the basic theory of orthogonal polynomials and a selection of exercises at the end of each chapter, it is suitable for graduate students with a solid background in stochastic processes as well as researchers in orthogonal polynomials and special functions who want to learn about applications of their work to probability.
Author | : R. D. Cooper |
Publisher | : |
Total Pages | : 112 |
Release | : 1975 |
Genre | : Functions, Special |
ISBN | : |
Download Stochastic Processes and Special Functions Book in PDF, Epub and Kindle
Author | : Shai M. J. Haran |
Publisher | : Springer Science & Business Media |
Total Pages | : 224 |
Release | : 2008-04-25 |
Genre | : Mathematics |
ISBN | : 3540783784 |
Download Arithmetical Investigations Book in PDF, Epub and Kindle
In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp,w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.
Author | : Giovanni Peccati |
Publisher | : Springer Science & Business Media |
Total Pages | : 281 |
Release | : 2011-04-06 |
Genre | : Mathematics |
ISBN | : 8847016797 |
Download Wiener Chaos: Moments, Cumulants and Diagrams Book in PDF, Epub and Kindle
The concept of Wiener chaos generalizes to an infinite-dimensional setting the properties of orthogonal polynomials associated with probability distributions on the real line. It plays a crucial role in modern probability theory, with applications ranging from Malliavin calculus to stochastic differential equations and from probabilistic approximations to mathematical finance. This book is concerned with combinatorial structures arising from the study of chaotic random variables related to infinitely divisible random measures. The combinatorial structures involved are those of partitions of finite sets, over which Möbius functions and related inversion formulae are defined. This combinatorial standpoint (which is originally due to Rota and Wallstrom) provides an ideal framework for diagrams, which are graphical devices used to compute moments and cumulants of random variables. Several applications are described, in particular, recent limit theorems for chaotic random variables. An Appendix presents a computer implementation in MATHEMATICA for many of the formulae.
Author | : Percy Deift |
Publisher | : American Mathematical Soc. |
Total Pages | : 273 |
Release | : 2000 |
Genre | : Orthogonal polynomials |
ISBN | : 0821826956 |
Download Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach Book in PDF, Epub and Kindle
This volume expands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. The central question was the following: Why do very general ensembles of random n times n matrices exhibit universal behavior as n > infinity? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.
Author | : Eli Levin |
Publisher | : Springer |
Total Pages | : 170 |
Release | : 2018-02-13 |
Genre | : Mathematics |
ISBN | : 3319729470 |
Download Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights Book in PDF, Epub and Kindle
This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals. Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics. This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices.
Author | : K Farahmand |
Publisher | : CRC Press |
Total Pages | : 180 |
Release | : 1998-08-15 |
Genre | : Mathematics |
ISBN | : 9780582356221 |
Download Topics in Random Polynomials Book in PDF, Epub and Kindle
Topics in Random Polynomials presents a rigorous and comprehensive treatment of the mathematical behavior of different types of random polynomials. These polynomials-the subject of extensive recent research-have many applications in physics, economics, and statistics. The main results are presented in such a fashion that they can be understood and used by readers whose knowledge of probability incorporates little more than basic probability theory and stochastic processes.
Author | : Rainer Buckdahn |
Publisher | : CRC Press |
Total Pages | : 296 |
Release | : 2002-05-16 |
Genre | : Mathematics |
ISBN | : 1482265230 |
Download Stochastic Processes and Related Topics Book in PDF, Epub and Kindle
This volume comprises selected papers presented at the 12th Winter School on Stochastic Processes and their Applications, which was held in Siegmundsburg, Germany, in March 2000. The contents include Backward Stochastic Differential Equations; Semilinear PDE and SPDE; Arbitrage Theory; Credit Derivatives and Models for Correlated Defaults; Three In
Author | : Shai M. J. Haran |
Publisher | : Springer |
Total Pages | : 0 |
Release | : 2008-05-02 |
Genre | : Mathematics |
ISBN | : 9783540783787 |
Download Arithmetical Investigations Book in PDF, Epub and Kindle
In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp,w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.