Orthogonal Polynomials in the Spectral Analysis of Markov Processes
Title | Orthogonal Polynomials in the Spectral Analysis of Markov Processes PDF eBook |
Author | Manuel Domínguez de la Iglesia |
Publisher | Cambridge University Press |
Pages | 348 |
Release | 2021-10-21 |
Genre | Mathematics |
ISBN | 1009035207 |
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 Orthogonal Polynomials
Title | Stochastic Processes and Orthogonal Polynomials PDF eBook |
Author | Wim Schoutens |
Publisher | Springer Science & Business Media |
Pages | 170 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461211700 |
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.
Geometric Methods in Physics XL
Title | Geometric Methods in Physics XL PDF eBook |
Author | Piotr Kielanowski |
Publisher | Springer Nature |
Pages | 465 |
Release | |
Genre | |
ISBN | 3031624076 |
Stochastic Processes and Orthogonal Polynomials
Title | Stochastic Processes and Orthogonal Polynomials PDF eBook |
Author | Wim Schoutens |
Publisher | |
Pages | 186 |
Release | 2000-04-01 |
Genre | |
ISBN | 9781461211716 |
Orthogonal Polynomials on the Unit Circle: Spectral theory
Title | Orthogonal Polynomials on the Unit Circle: Spectral theory PDF eBook |
Author | Barry Simon |
Publisher | American Mathematical Soc. |
Pages | 608 |
Release | 2005 |
Genre | Mathematics |
ISBN | 9780821836750 |
Presents an overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. This book discusses topics such as asymptotics of Toeplitz determinants (Szego's theorems), and limit theorems for the density of the zeros of orthogonal polynomials.
Orthogonal Polynomials on the Unit Circle
Title | Orthogonal Polynomials on the Unit Circle PDF eBook |
Author | Barry Simon |
Publisher | American Mathematical Soc. |
Pages | 610 |
Release | 2005 |
Genre | Education |
ISBN | 082184864X |
This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line. The book is suitable for graduate students and researchers interested in analysis.
Compound Renewal Processes
Title | Compound Renewal Processes PDF eBook |
Author | A. A. Borovkov |
Publisher | Cambridge University Press |
Pages | |
Release | 2022-06-30 |
Genre | Mathematics |
ISBN | 100911560X |
Compound renewal processes (CRPs) are among the most ubiquitous models arising in applications of probability. At the same time, they are a natural generalization of random walks, the most well-studied classical objects in probability theory. This monograph, written for researchers and graduate students, presents the general asymptotic theory and generalizes many well-known results concerning random walks. The book contains the key limit theorems for CRPs, functional limit theorems, integro-local limit theorems, large and moderately large deviation principles for CRPs in the state space and in the space of trajectories, including large deviation principles in boundary crossing problems for CRPs, with an explicit form of the rate functionals, and an extension of the invariance principle for CRPs to the domain of moderately large and small deviations. Applications establish the key limit laws for Markov additive processes, including limit theorems in the domains of normal and large deviations.