Stochastic Processes And Orthogonal Polynomials

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Stochastic Processes and Orthogonal Polynomials

Author : Wim Schoutens
Publisher : Springer Science & Business Media
ISBN 13 : 1461211700
Total Pages : 170 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Stochastic Processes and Orthogonal Polynomials by : Wim Schoutens

Download or read book Stochastic Processes and Orthogonal Polynomials written by Wim Schoutens and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Orthogonal Polynomials in the Spectral Analysis of Markov Processes

Author : Manuel Domínguez de la Iglesia
Publisher : Cambridge University Press
ISBN 13 : 1009035207
Total Pages : 348 pages
Book Rating : 4.0/5 (9 download)

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Book Synopsis Orthogonal Polynomials in the Spectral Analysis of Markov Processes by : Manuel Domínguez de la Iglesia

Download or read book Orthogonal Polynomials in the Spectral Analysis of Markov Processes written by Manuel Domínguez de la Iglesia and published by Cambridge University Press. This book was released on 2021-10-21 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Stochastic Processes and Special Functions

Author : R. D. Cooper
Publisher :
ISBN 13 :
Total Pages : 112 pages
Book Rating : 4.:/5 (773 download)

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Book Synopsis Stochastic Processes and Special Functions by : R. D. Cooper

Download or read book Stochastic Processes and Special Functions written by R. D. Cooper and published by . This book was released on 1975 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Arithmetical Investigations

Author : Shai M. J. Haran
Publisher : Springer Science & Business Media
ISBN 13 : 3540783784
Total Pages : 224 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Arithmetical Investigations by : Shai M. J. Haran

Download or read book Arithmetical Investigations written by Shai M. J. Haran and published by Springer Science & Business Media. This book was released on 2008-04-25 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Wiener Chaos: Moments, Cumulants and Diagrams

Author : Giovanni Peccati
Publisher : Springer Science & Business Media
ISBN 13 : 8847016797
Total Pages : 281 pages
Book Rating : 4.8/5 (47 download)

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Book Synopsis Wiener Chaos: Moments, Cumulants and Diagrams by : Giovanni Peccati

Download or read book Wiener Chaos: Moments, Cumulants and Diagrams written by Giovanni Peccati and published by Springer Science & Business Media. This book was released on 2011-04-06 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Topics in Random Polynomials

Author : K Farahmand
Publisher : CRC Press
ISBN 13 : 9780582356221
Total Pages : 180 pages
Book Rating : 4.3/5 (562 download)

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Book Synopsis Topics in Random Polynomials by : K Farahmand

Download or read book Topics in Random Polynomials written by K Farahmand and published by CRC Press. This book was released on 1998-08-15 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights

Author : Eli Levin
Publisher : Springer
ISBN 13 : 3319729470
Total Pages : 168 pages
Book Rating : 4.3/5 (197 download)

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Book Synopsis Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights by : Eli Levin

Download or read book Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights written by Eli Levin and published by Springer. This book was released on 2018-02-13 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Arithmetical Investigations

Author : Shai M. J. Haran
Publisher : Springer
ISBN 13 : 9783540783787
Total Pages : 0 pages
Book Rating : 4.7/5 (837 download)

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Book Synopsis Arithmetical Investigations by : Shai M. J. Haran

Download or read book Arithmetical Investigations written by Shai M. J. Haran and published by Springer. This book was released on 2008-05-02 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Stochastic Processes

Author : Jyotiprasad Medhi
Publisher : New Age International
ISBN 13 : 9788122405491
Total Pages : 664 pages
Book Rating : 4.4/5 (54 download)

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Book Synopsis Stochastic Processes by : Jyotiprasad Medhi

Download or read book Stochastic Processes written by Jyotiprasad Medhi and published by New Age International. This book was released on 1994 with total page 664 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aims At The Level Between That Of Elementary Probability Texts And Advanced Works On Stochastic Processes. The Pre-Requisites Are A Course On Elementary Probability Theory And Statistics, And A Course On Advanced Calculus. The Theoretical Results Developed Have Been Followed By A Large Number Of Illustrative Examples. These Have Been Supplemented By Numerous Exercises, Answers To Most Of Which Are Also Given. It Will Suit As A Text For Advanced Undergraduate, Postgraduate And Research Level Course In Applied Mathematics, Statistics, Operations Research, Computer Science, Different Branches Of Engineering, Telecommunications, Business And Management, Economics, Life Sciences And So On. A Review Of The Book In American Mathematical Monthly (December 82) Gives This Book Special Positive Emphasis As A Textbook As Follows: 'Of The Dozen Or More Texts Published In The Last Five Years Aimed At The Students With A Background Of A First Course In Probability And Statistics But Not Yet To Measure Theory, This Is The Clear Choice. An Extremely Well Organized, Lucidly Written Text With Numerous Problems, Examples And Reference T* (With T* Where T Denotes Textbook And * Denotes Special Positive Emphasis). The Current Enlarged And Revised Edition, While Retaining The Structure And Adhering To The Objective As Well As Philosophy Of The Earlier Edition, Removes The Deficiencies, Updates The Material And The References And Aims At A Border Perspective With Substantial Additions And Wider Coverage.

The Theory of Stochastic Processes I

Author : Iosif I. Gikhman
Publisher : Springer
ISBN 13 : 3642619436
Total Pages : 587 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis The Theory of Stochastic Processes I by : Iosif I. Gikhman

Download or read book The Theory of Stochastic Processes I written by Iosif I. Gikhman and published by Springer. This book was released on 2015-03-30 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the Reviews: "Gihman and Skorohod have done an excellent job of presenting the theory in its present state of rich imperfection." --D.W. Stroock, Bulletin of the American Mathematical Society, 1980

Stochastic Monotonicity and Queueing Applications of Birth-Death Processes

Author : Erik van Doorn
Publisher : Springer Science & Business Media
ISBN 13 : 1461258839
Total Pages : 125 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Stochastic Monotonicity and Queueing Applications of Birth-Death Processes by : Erik van Doorn

Download or read book Stochastic Monotonicity and Queueing Applications of Birth-Death Processes written by Erik van Doorn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: A stochastic process {X(t): 0 S t =} with discrete state space S c ~ is said to be stochastically increasing (decreasing) on an interval T if the probabilities Pr{X(t) i}, i E S, are increasing (decreasing) with t on T. Stochastic monotonicity is a basic structural property for process behaviour. It gives rise to meaningful bounds for various quantities such as the moments of the process, and provides the mathematical groundwork for approximation algorithms. Obviously, stochastic monotonicity becomes a more tractable subject for analysis if the processes under consideration are such that stochastic mono tonicity on an inter val 0

The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations

Author :
Publisher :
ISBN 13 :
Total Pages : 27 pages
Book Rating : 4.:/5 (227 download)

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Book Synopsis The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations by :

Download or read book The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations written by and published by . This book was released on 2003 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a new method for solving stochastic differential equations based on Galerking projections and extensions of Wiener's polynomial chaos. Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error. Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte-Carlo simulations for low dimensional stochastic inputs.

Stochastic Processes

Author : Andrei N Borodin
Publisher : Birkhäuser
ISBN 13 : 3319623109
Total Pages : 641 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis Stochastic Processes by : Andrei N Borodin

Download or read book Stochastic Processes written by Andrei N Borodin and published by Birkhäuser. This book was released on 2017-10-30 with total page 641 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a rigorous yet accessible introduction to the theory of stochastic processes. A significant part of the book is devoted to the classic theory of stochastic processes. In turn, it also presents proofs of well-known results, sometimes together with new approaches. Moreover, the book explores topics not previously covered elsewhere, such as distributions of functionals of diffusions stopped at different random times, the Brownian local time, diffusions with jumps, and an invariance principle for random walks and local times. Supported by carefully selected material, the book showcases a wealth of examples that demonstrate how to solve concrete problems by applying theoretical results. It addresses a broad range of applications, focusing on concrete computational techniques rather than on abstract theory. The content presented here is largely self-contained, making it suitable for researchers and graduate students alike.

Computational Aspects of Linear Control

Author : Claude Brezinski
Publisher : Springer Science & Business Media
ISBN 13 : 1461302617
Total Pages : 296 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Computational Aspects of Linear Control by : Claude Brezinski

Download or read book Computational Aspects of Linear Control written by Claude Brezinski and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many devices (we say dynamical systems or simply systems) behave like black boxes: they receive an input, this input is transformed following some laws (usually a differential equation) and an output is observed. The problem is to regulate the input in order to control the output, that is for obtaining a desired output. Such a mechanism, where the input is modified according to the output measured, is called feedback. The study and design of such automatic processes is called control theory. As we will see, the term system embraces any device and control theory has a wide variety of applications in the real world. Control theory is an interdisci plinary domain at the junction of differential and difference equations, system theory and statistics. Moreover, the solution of a control problem involves many topics of numerical analysis and leads to many interesting computational problems: linear algebra (QR, SVD, projections, Schur complement, structured matrices, localization of eigenvalues, computation of the rank, Jordan normal form, Sylvester and other equations, systems of linear equations, regulariza tion, etc), root localization for polynomials, inversion of the Laplace transform, computation of the matrix exponential, approximation theory (orthogonal poly nomials, Pad6 approximation, continued fractions and linear fractional transfor mations), optimization, least squares, dynamic programming, etc. So, control theory is also a. good excuse for presenting various (sometimes unrelated) issues of numerical analysis and the procedures for their solution. This book is not a book on control.

Asymptotics for Orthogonal Polynomials

Author : Walter Van Assche
Publisher : Lecture Notes in Mathematics
ISBN 13 :
Total Pages : 218 pages
Book Rating : 4.F/5 ( download)

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Book Synopsis Asymptotics for Orthogonal Polynomials by : Walter Van Assche

Download or read book Asymptotics for Orthogonal Polynomials written by Walter Van Assche and published by Lecture Notes in Mathematics. This book was released on 1987-06-24 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recently there has been a great deal of interest in the theory of orthogonal polynomials. The number of books treating the subject, however, is limited. This monograph brings together some results involving the asymptotic behaviour of orthogonal polynomials when the degree tends to infinity, assuming only a basic knowledge of real and complex analysis. An extensive treatment, starting with special knowledge of the orthogonality measure, is given for orthogonal polynomials on a compact set and on an unbounded set. Another possible approach is to start from properties of the coefficients in the three-term recurrence relation for orthogonal polynomials. This is done using the methods of (discrete) scattering theory. A new method, based on limit theorems in probability theory, to obtain asymptotic formulas for some polynomials is also given. Various consequences of all the results are described and applications are given ranging from random matrices and birth-death processes to discrete Schrödinger operators, illustrating the close interaction with different branches of applied mathematics.

Extrapolation and Rational Approximation

Author : Claude Brezinski
Publisher : Springer Nature
ISBN 13 : 3030584186
Total Pages : 410 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Extrapolation and Rational Approximation by : Claude Brezinski

Download or read book Extrapolation and Rational Approximation written by Claude Brezinski and published by Springer Nature. This book was released on 2020-11-30 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book paints a fresco of the field of extrapolation and rational approximation over the last several centuries to the present through the works of their primary contributors. It can serve as an introduction to the topics covered, including extrapolation methods, Padé approximation, orthogonal polynomials, continued fractions, Lanczos-type methods etc.; it also provides in depth discussion of the many links between these subjects. A highlight of this book is the presentation of the human side of the fields discussed via personal testimonies from contemporary researchers, their anecdotes, and their exclusive remembrances of some of the “actors.” This book shows how research in this domain started and evolved. Biographies of other scholars encountered have also been included. An important branch of mathematics is described in its historical context, opening the way to new developments. After a mathematical introduction, the book contains a precise description of the mathematical landscape of these fields spanning from the 19th century to the first part of the 20th. After an analysis of the works produced after that period (in particular those of Richardson, Aitken, Shanks, Wynn, and others), the most recent developments and applications are reviewed.

Numerical Methods for Stochastic Computations

Author : Dongbin Xiu
Publisher : Princeton University Press
ISBN 13 : 1400835348
Total Pages : 142 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Numerical Methods for Stochastic Computations by : Dongbin Xiu

Download or read book Numerical Methods for Stochastic Computations written by Dongbin Xiu and published by Princeton University Press. This book was released on 2010-07-01 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: The@ first graduate-level textbook to focus on fundamental aspects of numerical methods for stochastic computations, this book describes the class of numerical methods based on generalized polynomial chaos (gPC). These fast, efficient, and accurate methods are an extension of the classical spectral methods of high-dimensional random spaces. Designed to simulate complex systems subject to random inputs, these methods are widely used in many areas of computer science and engineering. The book introduces polynomial approximation theory and probability theory; describes the basic theory of gPC methods through numerical examples and rigorous development; details the procedure for converting stochastic equations into deterministic ones; using both the Galerkin and collocation approaches; and discusses the distinct differences and challenges arising from high-dimensional problems. The last section is devoted to the application of gPC methods to critical areas such as inverse problems and data assimilation. Ideal for use by graduate students and researchers both in the classroom and for self-study, Numerical Methods for Stochastic Computations provides the required tools for in-depth research related to stochastic computations. The first graduate-level textbook to focus on the fundamentals of numerical methods for stochastic computations Ideal introduction for graduate courses or self-study Fast, efficient, and accurate numerical methods Polynomial approximation theory and probability theory included Basic gPC methods illustrated through examples