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An Introduction to Measure-Theoretic Probability

An Introduction to Measure-Theoretic Probability
A Book

by George G. Roussas

  • Publisher : Academic Press
  • Release : 2014-03-19
  • Pages : 426
  • ISBN : 0128002905
  • Language : En, Es, Fr & De
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An Introduction to Measure-Theoretic Probability, Second Edition, employs a classical approach to teaching the basics of measure theoretic probability. This book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics, probability and other related areas should be equipped with. This edition requires no prior knowledge of measure theory, covers all its topics in great detail, and includes one chapter on the basics of ergodic theory and one chapter on two cases of statistical estimation. Topics range from the basic properties of a measure to modes of convergence of a sequence of random variables and their relationships; the integral of a random variable and its basic properties; standard convergence theorems; standard moment and probability inequalities; the Hahn-Jordan Decomposition Theorem; the Lebesgue Decomposition T; conditional expectation and conditional probability; theory of characteristic functions; sequences of independent random variables; and ergodic theory. There is a considerable bend toward the way probability is actually used in statistical research, finance, and other academic and nonacademic applied pursuits. Extensive exercises and practical examples are included, and all proofs are presented in full detail. Complete and detailed solutions to all exercises are available to the instructors on the book companion site. This text will be a valuable resource for graduate students primarily in statistics, mathematics, electrical and computer engineering or other information sciences, as well as for those in mathematical economics/finance in the departments of economics. Provides in a concise, yet detailed way, the bulk of probabilistic tools essential to a student working toward an advanced degree in statistics, probability, and other related fields Includes extensive exercises and practical examples to make complex ideas of advanced probability accessible to graduate students in statistics, probability, and related fields All proofs presented in full detail and complete and detailed solutions to all exercises are available to the instructors on book companion site Considerable bend toward the way probability is used in statistics in non-mathematical settings in academic, research and corporate/finance pursuits.

Introduction to Measure-Theoretic Probability Instructors Manual

Introduction to Measure-Theoretic Probability Instructors Manual
A Book

by Elsevier Science & Technology

  • Publisher : Academic Press
  • Release : 2004-11
  • Pages : 598
  • ISBN : 9780120883899
  • Language : En, Es, Fr & De
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Measure Theory

Measure Theory
Second Edition

by Donald L. Cohn

  • Publisher : Springer Science & Business Media
  • Release : 2013-07-13
  • Pages : 457
  • ISBN : 1461469562
  • Language : En, Es, Fr & De
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Intended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups. This second edition includes a chapter on measure-theoretic probability theory, plus brief treatments of the Banach-Tarski paradox, the Henstock-Kurzweil integral, the Daniell integral, and the existence of liftings. Measure Theory provides a solid background for study in both functional analysis and probability theory and is an excellent resource for advanced undergraduate and graduate students in mathematics. The prerequisites for this book are basic courses in point-set topology and in analysis, and the appendices present a thorough review of essential background material.

An Introduction to Econometric Theory

An Introduction to Econometric Theory
Measure-Theoretic Probability and Statistics with Applications to Economics

by A. Ronald Gallant

  • Publisher : Princeton University Press
  • Release : 2018-06-05
  • Pages : 129
  • ISBN : 0691186235
  • Language : En, Es, Fr & De
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Intended primarily to prepare first-year graduate students for their ongoing work in econometrics, economic theory, and finance, this innovative book presents the fundamental concepts of theoretical econometrics, from measure-theoretic probability to statistics. A. Ronald Gallant covers these topics at an introductory level and develops the ideas to the point where they can be applied. He thereby provides the reader not only with a basic grasp of the key empirical tools but with sound intuition as well. In addition to covering the basic tools of empirical work in economics and finance, Gallant devotes particular attention to motivating ideas and presenting them as the solution to practical problems. For example, he presents correlation, regression, and conditional expectation as a means of obtaining the best approximation of one random variable by some function of another. He considers linear, polynomial, and unrestricted functions, and leads the reader to the notion of conditioning on a sigma-algebra as a means for finding the unrestricted solution. The reader thus gains an understanding of the relationships among linear, polynomial, and unrestricted solutions. Proofs of results are presented when the proof itself aids understanding or when the proof technique has practical value. A major text-treatise by one of the leading scholars in this field, An Introduction to Econometric Theory will prove valuable not only to graduate students but also to all economists, statisticians, and finance professionals interested in the ideas and implications of theoretical econometrics.

Introduction to Probability Theory: A First Course on the Measure-Theoretic Approach

Introduction to Probability Theory: A First Course on the Measure-Theoretic Approach
A Book

by Nima Moshayedi

  • Publisher : Unknown Publisher
  • Release : 2022-04-08
  • Pages : 129
  • ISBN : 9789811246746
  • Language : En, Es, Fr & De
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This book provides a first introduction to the methods of probability theory by using the modern and rigorous techniques of measure theory and functional analysis. It is geared for undergraduate students, mainly in mathematics and physics majors, but also for students from other subject areas such as economics, finance and engineering. It is an invaluable source, either for a parallel use to a related lecture or for its own purpose of learning it.The first part of the book gives a basic introduction to probability theory. It explains the notions of random events and random variables, probability measures, expectation values, distributions, characteristic functions, independence of random variables, as well as different types of convergence and limit theorems. The first part contains two chapters. The first chapter presents combinatorial aspects of probability theory, and the second chapter delves into the actual introduction to probability theory, which contains the modern probability language. The second part is devoted to some more sophisticated methods such as conditional expectations, martingales and Markov chains. These notions will be fairly accessible after reading the first part.

A First Look at Rigorous Probability Theory

A First Look at Rigorous Probability Theory
A Book

by Jeffrey S. Rosenthal

  • Publisher : World Scientific Publishing Company Incorporated
  • Release : 2006-01-01
  • Pages : 219
  • ISBN : 9789812703712
  • Language : En, Es, Fr & De
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Features an introduction to probability theory using measure theory. This work provides proofs of the essential introductory results and presents the measure theory and mathematical details in terms of intuitive probabilistic concepts, rather than as separate, imposing subjects.

Probability for Statisticians

Probability for Statisticians
A Book

by Galen R. Shorack

  • Publisher : Springer
  • Release : 2017-10-03
  • Pages : 510
  • ISBN : 9783319522067
  • Language : En, Es, Fr & De
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The choice of examples used in this text clearly illustrate its use for a one-year graduate course. The material to be presented in the classroom constitutes a little more than half the text, while the rest of the text provides background, offers different routes that could be pursued in the classroom, as well as additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Steins method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function, with both the bootstrap and trimming presented. The section on martingales covers censored data martingales.

Measure, Integral and Probability

Measure, Integral and Probability
A Book

by Marek Capinski,Peter E. Kopp

  • Publisher : Springer Science & Business Media
  • Release : 2013-12-01
  • Pages : 311
  • ISBN : 1447106458
  • Language : En, Es, Fr & De
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Measure, Integral and Probability is a gentle introduction that makes measure and integration theory accessible to the average third-year undergraduate student. The ideas are developed at an easy pace in a form that is suitable for self-study, with an emphasis on clear explanations and concrete examples rather than abstract theory. For this second edition, the text has been thoroughly revised and expanded. New features include: · a substantial new chapter, featuring a constructive proof of the Radon-Nikodym theorem, an analysis of the structure of Lebesgue-Stieltjes measures, the Hahn-Jordan decomposition, and a brief introduction to martingales · key aspects of financial modelling, including the Black-Scholes formula, discussed briefly from a measure-theoretical perspective to help the reader understand the underlying mathematical framework. In addition, further exercises and examples are provided to encourage the reader to become directly involved with the material.

Probability, Random Processes, and Ergodic Properties

Probability, Random Processes, and Ergodic Properties
A Book

by Robert M. Gray

  • Publisher : Springer Science & Business Media
  • Release : 2009-07-31
  • Pages : 322
  • ISBN : 1441910905
  • Language : En, Es, Fr & De
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Probability, Random Processes, and Ergodic Properties is for mathematically inclined information/communication theorists and people working in signal processing. It will also interest those working with random or stochastic processes, including mathematicians, statisticians, and economists. Highlights: Complete tour of book and guidelines for use given in Introduction, so readers can see at a glance the topics of interest. Structures mathematics for an engineering audience, with emphasis on engineering applications. New in the Second Edition: Much of the material has been rearranged and revised for pedagogical reasons. The original first chapter has been split in order to allow a more thorough treatment of basic probability before tackling random processes and dynamical systems. The final chapter has been broken into two pieces to provide separate emphasis on process metrics and the ergodic decomposition of affine functionals. Many classic inequalities are now incorporated into the text, along with proofs; and many citations have been added.

Probability on Compact Lie Groups

Probability on Compact Lie Groups
A Book

by David Applebaum

  • Publisher : Springer
  • Release : 2014-06-26
  • Pages : 217
  • ISBN : 3319078429
  • Language : En, Es, Fr & De
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Probability theory on compact Lie groups deals with the interaction between “chance” and “symmetry,” a beautiful area of mathematics of great interest in its own sake but which is now also finding increasing applications in statistics and engineering (particularly with respect to signal processing). The author gives a comprehensive introduction to some of the principle areas of study, with an emphasis on applicability. The most important topics presented are: the study of measures via the non-commutative Fourier transform, existence and regularity of densities, properties of random walks and convolution semigroups of measures and the statistical problem of deconvolution. The emphasis on compact (rather than general) Lie groups helps readers to get acquainted with what is widely seen as a difficult field but which is also justified by the wealth of interesting results at this level and the importance of these groups for applications. The book is primarily aimed at researchers working in probability, stochastic analysis and harmonic analysis on groups. It will also be of interest to mathematicians working in Lie theory and physicists, statisticians and engineers who are working on related applications. A background in first year graduate level measure theoretic probability and functional analysis is essential; a background in Lie groups and representation theory is certainly helpful but the first two chapters also offer orientation in these subjects.

A User's Guide to Measure Theoretic Probability

A User's Guide to Measure Theoretic Probability
A Book

by David Pollard

  • Publisher : Cambridge University Press
  • Release : 2002
  • Pages : 351
  • ISBN : 9780521002899
  • Language : En, Es, Fr & De
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This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean.

Introduction to Imprecise Probabilities

Introduction to Imprecise Probabilities
A Book

by Thomas Augustin,Frank P. A. Coolen,Gert de Cooman,Matthias C. M. Troffaes

  • Publisher : John Wiley & Sons
  • Release : 2014-04-11
  • Pages : 448
  • ISBN : 1118763149
  • Language : En, Es, Fr & De
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In recent years, the theory has become widely accepted and has beenfurther developed, but a detailed introduction is needed in orderto make the material available and accessible to a wide audience.This will be the first book providing such an introduction,covering core theory and recent developments which can be appliedto many application areas. All authors of individual chapters areleading researchers on the specific topics, assuring high qualityand up-to-date contents. An Introduction to Imprecise Probabilities provides acomprehensive introduction to imprecise probabilities, includingtheory and applications reflecting the current state if the art.Each chapter is written by experts on the respective topics,including: Sets of desirable gambles; Coherent lower (conditional)previsions; Special cases and links to literature; Decision making;Graphical models; Classification; Reliability and risk assessment;Statistical inference; Structural judgments; Aspects ofimplementation (including elicitation and computation); Models infinance; Game-theoretic probability; Stochastic processes(including Markov chains); Engineering applications. Essential reading for researchers in academia, researchinstitutes and other organizations, as well as practitionersengaged in areas such as risk analysis and engineering.

Introdction to Measure and Probability

Introdction to Measure and Probability
A Book

by J. F. C. Kingman,S. J. Taylor

  • Publisher : Cambridge University Press
  • Release : 2008-11-20
  • Pages : 129
  • ISBN : 1316582159
  • Language : En, Es, Fr & De
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The authors believe that a proper treatment of probability theory requires an adequate background in the theory of finite measures in general spaces. The first part of their book sets out this material in a form that not only provides an introduction for intending specialists in measure theory but also meets the needs of students of probability. The theory of measure and integration is presented for general spaces, with Lebesgue measure and the Lebesgue integral considered as important examples whose special properties are obtained. The introduction to functional analysis which follows covers the material (such as the various notions of convergence) which is relevant to probability theory and also the basic theory of L2-spaces, important in modern physics. The second part of the book is an account of the fundamental theoretical ideas which underlie the applications of probability in statistics and elsewhere, developed from the results obtained in the first part. A large number of examples is included; these form an essential part of the development.

Measure, Probability, and Mathematical Finance

Measure, Probability, and Mathematical Finance
A Problem-Oriented Approach

by Guojun Gan,Chaoqun Ma,Hong Xie

  • Publisher : John Wiley & Sons
  • Release : 2014-04-07
  • Pages : 744
  • ISBN : 1118831969
  • Language : En, Es, Fr & De
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An introduction to the mathematical theory and financial models developed and used on Wall Street Providing both a theoretical and practical approach to the underlying mathematical theory behind financial models, Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach presents important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus. Measure theory is indispensable to the rigorous development of probability theory and is also necessary to properly address martingale measures, the change of numeraire theory, and LIBOR market models. In addition, probability theory is presented to facilitate the development of stochastic processes, including martingales and Brownian motions, while stochastic processes and stochastic calculus are discussed to model asset prices and develop derivative pricing models. The authors promote a problem-solving approach when applying mathematics in real-world situations, and readers are encouraged to address theorems and problems with mathematical rigor. In addition, Measure, Probability, and Mathematical Finance features: A comprehensive list of concepts and theorems from measure theory, probability theory, stochastic processes, and stochastic calculus Over 500 problems with hints and select solutions to reinforce basic concepts and important theorems Classic derivative pricing models in mathematical finance that have been developed and published since the seminal work of Black and Scholes Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach is an ideal textbook for introductory quantitative courses in business, economics, and mathematical finance at the upper-undergraduate and graduate levels. The book is also a useful reference for readers who need to build their mathematical skills in order to better understand the mathematical theory of derivative pricing models.

Introduction to the Mathematical and Statistical Foundations of Econometrics

Introduction to the Mathematical and Statistical Foundations of Econometrics
A Book

by Herman J. Bierens

  • Publisher : Cambridge University Press
  • Release : 2004-12-20
  • Pages : 323
  • ISBN : 9780521542241
  • Language : En, Es, Fr & De
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This book is intended for use in a rigorous introductory PhD level course in econometrics.

Measure, Integral, Probability & Processes

Measure, Integral, Probability & Processes
A Concise Introduction to Probability and Random Processes. Probab(ilistical)ly the Theoretical Minimum

by René L Schilling

  • Publisher : Unknown Publisher
  • Release : 2021-02-02
  • Pages : 450
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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In these lecture notes we give a self-contained and concise introduction to the essentials of modern probability theory. The material covers all concepts and techniques usually taught at BSc and first-year graduate level probability courses: Measure & integration theory, elementary probability theory, further probability, classic limit theorems, discrete-time and continuous-time martingales, Poisson processes, random walks & Markov chains and, finally, first steps towards Brownian motion. The text can serve as a course companion, for self study or as a reference text. Concepts, which will be useful for later chapters and further studies are introduced early on. The material is organized and presented in a way that will enable the readers to continue their study with any advanced text in probability theory, stochastic processes or stochastic analysis. Much emphasis is put on being reader-friendly and useful, giving a direct and quick start into a fascinating mathematical topic.

Applied Probability

Applied Probability
From Random Sequences to Stochastic Processes

by Valérie Girardin,Nikolaos Limnios

  • Publisher : Springer
  • Release : 2018-09-12
  • Pages : 260
  • ISBN : 3319974122
  • Language : En, Es, Fr & De
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This textbook addresses postgraduate students in applied mathematics, probability, and statistics, as well as computer scientists, biologists, physicists and economists, who are seeking a rigorous introduction to applied stochastic processes. Pursuing a pedagogic approach, the content follows a path of increasing complexity, from the simplest random sequences to the advanced stochastic processes. Illustrations are provided from many applied fields, together with connections to ergodic theory, information theory, reliability and insurance. The main content is also complemented by a wealth of examples and exercises with solutions.

Probability

Probability
Theory and Examples

by Rick Durrett

  • Publisher : Cambridge University Press
  • Release : 2019-04-30
  • Pages : 432
  • ISBN : 1108473687
  • Language : En, Es, Fr & De
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A well-written and lively introduction to measure theoretic probability for graduate students and researchers.

An Introduction to Sequential Monte Carlo

An Introduction to Sequential Monte Carlo
A Book

by Nicolas Chopin,Omiros Papaspiliopoulos

  • Publisher : Springer Nature
  • Release : 2020-10-01
  • Pages : 378
  • ISBN : 3030478459
  • Language : En, Es, Fr & De
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This book provides a general introduction to Sequential Monte Carlo (SMC) methods, also known as particle filters. These methods have become a staple for the sequential analysis of data in such diverse fields as signal processing, epidemiology, machine learning, population ecology, quantitative finance, and robotics. The coverage is comprehensive, ranging from the underlying theory to computational implementation, methodology, and diverse applications in various areas of science. This is achieved by describing SMC algorithms as particular cases of a general framework, which involves concepts such as Feynman-Kac distributions, and tools such as importance sampling and resampling. This general framework is used consistently throughout the book. Extensive coverage is provided on sequential learning (filtering, smoothing) of state-space (hidden Markov) models, as this remains an important application of SMC methods. More recent applications, such as parameter estimation of these models (through e.g. particle Markov chain Monte Carlo techniques) and the simulation of challenging probability distributions (in e.g. Bayesian inference or rare-event problems), are also discussed. The book may be used either as a graduate text on Sequential Monte Carlo methods and state-space modeling, or as a general reference work on the area. Each chapter includes a set of exercises for self-study, a comprehensive bibliography, and a “Python corner,” which discusses the practical implementation of the methods covered. In addition, the book comes with an open source Python library, which implements all the algorithms described in the book, and contains all the programs that were used to perform the numerical experiments.

Perspectives on Statistical Thermodynamics

Perspectives on Statistical Thermodynamics
A Book

by Yoshitsugu Oono

  • Publisher : Cambridge University Press
  • Release : 2017-11-30
  • Pages : 458
  • ISBN : 1107154014
  • Language : En, Es, Fr & De
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This original text develops a deep, conceptual understanding of thermal physics and highlights the important links between statistical physics and classical thermodynamics. It examines how thermal physics fits within physics as a whole, and is perfect for undergraduate and graduate students, and researchers interested in a fresh approach to the subject.