Download or read book Matrix Gamma Distributions and Related Stochastic Processes written by Tomasz J. Kozubowski and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is considerable literature on matrix-variate gamma distributions, also known as Wishart distributions, which are driven by a shape parameter with values in the (Gindikin) set {i/2, i = 1, . . . , k-1}∪((k-1)/2, ∞). We provide an extension of this class to the case where the shape parameter may actually take on any positive value. In addition to the well-known singular Wishart as well as non-singular matrix-variate gamma distributions, the proposed class includes new singular matrix-variate distributions, with the shape parameter outside of the Gindikin set. This singular, non-Wishart case is no longer permutation invariant and derivation of its scaling properties requires special care. Among numerous newly established properties of the extended class are group-like relations with respect to the positive shape parameter. The latter provide a natural substitute for the classical convolution properties that are crucial in the study of infinite divisibility. Our results provide further clarification regarding the lack of infinite divisibility of Wishart distributions, a classical observation of Paul L'evy. In particular, we clarify why the row/column vectors in the off-diagonal blocks are infinitely divisible. A class of matrix-variate Laplace distributions arises naturally in this set-up as the distributions of the off-diagonal blocks of random gamma matrices. For the class of Laplace rectangular matrices, we obtain distributional identities that follow from the role they play in the structure of the matrix gamma distributions. We present several elegant and convenient stochastic representations of the discussed classes of matrix-valued distributions. In particular, we show that the matrix-variate gamma distribution is a symmetrization of the triangular Rayleigh distributed matrix - a new class of the matrix variables that naturally extend the classical univariate Rayleigh variables. Finally, a connection of the matrix-variate gamma distributions to matrix-valued L'evy processes of a vector argument is made. Namely, a L'evy process, termed a matrix gammaLaplace motion, is obtained by the subordination of the triangular Brownian motion of a vector argument to a vector-valued gamma motion of a vector argument. In this context, we introduce a triangular matrix-valued Rayleigh process, which, through symmetrization, leads to a new matrix-variate gamma process. This process when taken at a properly defined one-dimensional argument has the matrix gamma marginal distribution with the shape parameter equal to its argument.

Download or read book Student s t Distribution and Related Stochastic Processes written by Bronius Grigelionis and published by Springer Science & Business Media. This book was released on 2012-09-18 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: This brief monograph is an in-depth study of the infinite divisibility and self-decomposability properties of central and noncentral Student’s distributions, represented as variance and mean-variance mixtures of multivariate Gaussian distributions with the reciprocal gamma mixing distribution. These results allow us to define and analyse Student-Lévy processes as Thorin subordinated Gaussian Lévy processes. A broad class of one-dimensional, strictly stationary diffusions with the Student’s t-marginal distribution are defined as the unique weak solution for the stochastic differential equation. Using the independently scattered random measures generated by the bi-variate centred Student-Lévy process, and stochastic integration theory, a univariate, strictly stationary process with the centred Student’s t- marginals and the arbitrary correlation structure are defined. As a promising direction for future work in constructing and analysing new multivariate Student-Lévy type processes, the notion of Lévy copulas and the related analogue of Sklar’s theorem are explained.

Download or read book Stochastic Processes written by M. Girault and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: Existing works on stochastic processes belong to a field of abstract mathematics which puts them beyond the scope of the non-specialist. The preoccupations of research mathematicians being more often than not distant from the practical problems of experimental methodology, the needs of practical workers, though real, are not met by the majority of works that. deal with processes. By "practical workers", we mean research scientists in all the different disciplines: Physics, Chemistry, Biology, Medicine, Population, Economics, Organisation, Operational Research etc. Indeed, all scientific research today touches upon complex fields in which deterministic models can be useful for no more than an element ary and simple approximation. The Calculus of Probability although offering some interesting models is still inadequate in many instances, particularly in the study of evolving systems. The practical worker must therefore have at his disposal a set of original and varied stochastic models. These models must not be too general, for in that case not only would their theoretical study prove difficult, but above all the adaptation of such models to an observed system would lead to an estimation of a great number of parameters on the basis of a necessarily restricted sample. This would constitute an insuperable difficulty for the practical scientist. It is therefore essential for him to have at his disposal a varied range of very characteristic models.

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.

Download or read book Matrix Exponential Distributions in Applied Probability written by Mogens Bladt and published by Springer. This book was released on 2017-05-18 with total page 749 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains an in-depth treatment of matrix-exponential (ME) distributions and their sub-class of phase-type (PH) distributions. Loosely speaking, an ME distribution is obtained through replacing the intensity parameter in an exponential distribution by a matrix. The ME distributions can also be identified as the class of non-negative distributions with rational Laplace transforms. If the matrix has the structure of a sub-intensity matrix for a Markov jump process we obtain a PH distribution which allows for nice probabilistic interpretations facilitating the derivation of exact solutions and closed form formulas. The full potential of ME and PH unfolds in their use in stochastic modelling. Several chapters on generic applications, like renewal theory, random walks and regenerative processes, are included together with some specific examples from queueing theory and insurance risk. We emphasize our intention towards applications by including an extensive treatment on statistical methods for PH distributions and related processes that will allow practitioners to calibrate models to real data. Aimed as a textbook for graduate students in applied probability and statistics, the book provides all the necessary background on Poisson processes, Markov chains, jump processes, martingales and re-generative methods. It is our hope that the provided background may encourage researchers and practitioners from other fields, like biology, genetics and medicine, who wish to become acquainted with the matrix-exponential method and its applications.

Download or read book Statistical Analysis of Stochastic Processes in Time written by J. K. Lindsey and published by Cambridge University Press. This book was released on 2004-08-02 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book was first published in 2004. Many observed phenomena, from the changing health of a patient to values on the stock market, are characterised by quantities that vary over time: stochastic processes are designed to study them. This book introduces practical methods of applying stochastic processes to an audience knowledgeable only in basic statistics. It covers almost all aspects of the subject and presents the theory in an easily accessible form that is highlighted by application to many examples. These examples arise from dozens of areas, from sociology through medicine to engineering. Complementing these are exercise sets making the book suited for introductory courses in stochastic processes. Software (available from www.cambridge.org) is provided for the freely available R system for the reader to apply to all the models presented.

Download or read book Basic Stochastic Processes written by Pierre Devolder and published by John Wiley & Sons. This book was released on 2015-08-05 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents basic stochastic processes, stochastic calculus including Lévy processes on one hand, and Markov and Semi Markov models on the other. From the financial point of view, essential concepts such as the Black and Scholes model, VaR indicators, actuarial evaluation, market values, fair pricing play a central role and will be presented. The authors also present basic concepts so that this series is relatively self-contained for the main audience formed by actuaries and particularly with ERM (enterprise risk management) certificates, insurance risk managers, students in Master in mathematics or economics and people involved in Solvency II for insurance companies and in Basel II and III for banks.

Download or read book Stochastic Processes written by Peter Watts Jones and published by CRC Press. This book was released on 2009-10-09 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on a highly popular, well-established course taught by the authors, Stochastic Processes: An Introduction, Second Edition discusses the modeling and analysis of random experiments using the theory of probability. It focuses on the way in which the results or outcomes of experiments vary and evolve over time. The text begins with a review of relevant fundamental probability. It then covers several basic gambling problems, random walks, and Markov chains. The authors go on to develop random processes continuous in time, including Poisson, birth and death processes, and general population models. While focusing on queues, they present an extended discussion on the analysis of associated stationary processes. The book also explores reliability and other random processes, such as branching processes, martingales, and a simple epidemic. The appendix contains key mathematical results for reference. Ideal for a one-semester course on stochastic processes, this concise, updated textbook makes the material accessible to students by avoiding specialized applications and instead highlighting simple applications and examples. The associated website contains Mathematica® and R programs that offer flexibility in creating graphs and performing computations.

Download or read book Stochastic Processes and Random Matrices written by Grégory Schehr and published by Oxford University Press. This book was released on 2017-08-15 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).

Download or read book Applied Stochastic Processes written by Mario Lefebvre and published by Springer Science & Business Media. This book was released on 2007-12-14 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book uses a distinctly applied framework to present the most important topics in stochastic processes, including Gaussian and Markovian processes, Markov Chains, Poisson processes, Brownian motion and queueing theory. The book also examines in detail special diffusion processes, with implications for finance, various generalizations of Poisson processes, and renewal processes. It contains numerous examples and approximately 350 advanced problems that reinforce both concepts and applications. Entertaining mini-biographies of mathematicians give an enriching historical context. The book includes statistical tables and solutions to the even-numbered problems at the end.

Download or read book The Analysis of Stochastic Processes using GLIM written by James K. Lindsey and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to present a survey of the many ways in which the statistical package GLIM may be used to model and analyze stochastic processes. Its emphasis is on using GLIM interactively to apply statistical techniques, and examples are drawn from a wide range of applications including medicine, biology, and the social sciences. It is based on the author's many years of teaching courses along these lines to both undergraduate and graduate students. The author assumes that readers have a reasonably strong background in statistics such as might be gained from undergraduate courses and that they are also familiar with the basic workings of GLIM. Topics covered include: the analysis of survival data, regression and fitting distributions, time series analysis (including both the time and frequency domains), repeated measurements, and generalized linear models.

Download or read book Introduction to Matrix Analytic Methods in Stochastic Modeling written by G. Latouche and published by SIAM. This book was released on 1999-01-01 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents the basic mathematical ideas and algorithms of the matrix analytic theory in a readable, up-to-date, and comprehensive manner.

Download or read book Statistical Distributions in Scientific Work written by Charles Taillie and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the NATO Advanced Study Institute, Trieste, Italy, July 10-August 1, 1980

Download or read book Generalized Gamma Convolutions and Related Classes of Distributions and Densities written by Lennart Bondesson and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: Generalized Gamma convolutions were introduced by Olof Thorin in 1977 and were used by him to show that, in particular, the Lognormal distribution is infinitely divisible. After that a large number of papers rapidly appeared with new results in a somewhat random order. Many of the papers appeared in the Scandinavian Actuarial Journal. This work is an attempt to present the main results on this class of probability distributions and related classes in a rather logical order. The goal has been to be on a level that is not too advanced. However, since the field is rather technical, most readers will find difficult passages in the text. Those who do not want to visit a mysterious land situated between the land of probability theory and statistics and the land of classical analysis should not look at this work. When some years ago I submitted a survey to a journal it was suggested by the editor, K. Krickeberg, that it should be expanded to a book. However, at that time I was rather reluctant to do so since there remained so many problems to be solved or to be solved in a smoother way than before. Moreover, there was at that time some lack of probabilistic interpretations and applications. Many of the problems are now solved but still it is felt that more applications than those presented in the work could be found.

Download or read book Random Matrices Random Processes and Integrable Systems written by John Harnad and published by Springer Science & Business Media. This book was released on 2011-05-06 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the remarkable connections between two domains that, a priori, seem unrelated: Random matrices (together with associated random processes) and integrable systems. The relations between random matrix models and the theory of classical integrable systems have long been studied. These appear mainly in the deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based upon the Riemann-Hilbert problem of analytic function theory and by related approaches to the study of nonlinear asymptotics in the large N limit. Associated with studies of matrix models are certain stochastic processes, the "Dyson processes", and their continuum diffusion limits, which govern the spectrum in random matrix ensembles, and may also be studied by related methods. Random Matrices, Random Processes and Integrable Systems provides an in-depth examination of random matrices with applications over a vast variety of domains, including multivariate statistics, random growth models, and many others. Leaders in the field apply the theory of integrable systems to the solution of fundamental problems in random systems and processes using an interdisciplinary approach that sheds new light on a dynamic topic of current research.

Download or read book Level Sets and Extrema of Random Processes and Fields written by Jean-Marc Azais and published by John Wiley & Sons. This book was released on 2009-02-17 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: A timely and comprehensive treatment of random field theory with applications across diverse areas of study Level Sets and Extrema of Random Processes and Fields discusses how to understand the properties of the level sets of paths as well as how to compute the probability distribution of its extremal values, which are two general classes of problems that arise in the study of random processes and fields and in related applications. This book provides a unified and accessible approach to these two topics and their relationship to classical theory and Gaussian processes and fields, and the most modern research findings are also discussed. The authors begin with an introduction to the basic concepts of stochastic processes, including a modern review of Gaussian fields and their classical inequalities. Subsequent chapters are devoted to Rice formulas, regularity properties, and recent results on the tails of the distribution of the maximum. Finally, applications of random fields to various areas of mathematics are provided, specifically to systems of random equations and condition numbers of random matrices. Throughout the book, applications are illustrated from various areas of study such as statistics, genomics, and oceanography while other results are relevant to econometrics, engineering, and mathematical physics. The presented material is reinforced by end-of-chapter exercises that range in varying degrees of difficulty. Most fundamental topics are addressed in the book, and an extensive, up-to-date bibliography directs readers to existing literature for further study. Level Sets and Extrema of Random Processes and Fields is an excellent book for courses on probability theory, spatial statistics, Gaussian fields, and probabilistic methods in real computation at the upper-undergraduate and graduate levels. It is also a valuable reference for professionals in mathematics and applied fields such as statistics, engineering, econometrics, mathematical physics, and biology.

Download or read book Bayesian Inference for Stochastic Processes written by Lyle D. Broemeling and published by CRC Press. This book was released on 2017-12-12 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book designed to introduce Bayesian inference procedures for stochastic processes. There are clear advantages to the Bayesian approach (including the optimal use of prior information). Initially, the book begins with a brief review of Bayesian inference and uses many examples relevant to the analysis of stochastic processes, including the four major types, namely those with discrete time and discrete state space and continuous time and continuous state space. The elements necessary to understanding stochastic processes are then introduced, followed by chapters devoted to the Bayesian analysis of such processes. It is important that a chapter devoted to the fundamental concepts in stochastic processes is included. Bayesian inference (estimation, testing hypotheses, and prediction) for discrete time Markov chains, for Markov jump processes, for normal processes (e.g. Brownian motion and the Ornstein–Uhlenbeck process), for traditional time series, and, lastly, for point and spatial processes are described in detail. Heavy emphasis is placed on many examples taken from biology and other scientific disciplines. In order analyses of stochastic processes, it will use R and WinBUGS. Features: Uses the Bayesian approach to make statistical Inferences about stochastic processes The R package is used to simulate realizations from different types of processes Based on realizations from stochastic processes, the WinBUGS package will provide the Bayesian analysis (estimation, testing hypotheses, and prediction) for the unknown parameters of stochastic processes To illustrate the Bayesian inference, many examples taken from biology, economics, and astronomy will reinforce the basic concepts of the subject A practical approach is implemented by considering realistic examples of interest to the scientific community WinBUGS and R code are provided in the text, allowing the reader to easily verify the results of the inferential procedures found in the many examples of the book Readers with a good background in two areas, probability theory and statistical inference, should be able to master the essential ideas of this book.