An Introduction to MeasureTheoretic Probability
2nd Edition
Description
An Introduction to MeasureTheoretic 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 HahnJordan 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.
Key Features
 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 nonmathematical settings in academic, research and corporate/finance pursuits.
Readership
Graduate students primarily in statistics, mathematics, electrical & computer engineering or other information sciences; mathematical economics/finance in departments of economics.
Table of Contents
 Dedication
 Pictured on the Cover
 Carathéodory, Constantine (1873–1950)
 Preface to First Edition
 Preface to Second Edition
 Chapter 1. Certain Classes of Sets, Measurability, and Pointwise Approximation
 Abstract
 1.1 Measurable Spaces
 1.2 Product Measurable Spaces
 1.3 Measurable Functions and Random Variables
 Chapter 2. Definition and Construction of a Measure and its Basic Properties
 Abstract
 2.1 About Measures in General, and Probability Measures in Particular
 2.2 Outer Measures
 2.3 The Carathéodory Extension Theorem
 2.4 Measures and (Point) Functions
 Chapter 3. Some Modes of Convergence of Sequences of Random Variables and their Relationships
 Abstract
 3.1 Almost Everywhere Convergence and Convergence in Measure
 3.2 Convergence in Measure is Equivalent to Mutual Convergence in Measure
 Chapter 4. The Integral of a Random Variable and its Basic Properties
 Abstract
 4.1 Definition of the Integral
 4.2 Basic Properties of the Integral
 4.3 Probability Distributions
 Chapter 5. Standard Convergence Theorems, The Fubini Theorem
 Abstract
 5.1 Standard Convergence Theorems and Some of Their Ramifications
 5.2 Sections, Product Measure Theorem, the Fubini Theorem
 Chapter 6. Standard Moment and Probability Inequalities, Convergence in the rth Mean and its Implications
 Abstract
 6.1 Moment and Probability Inequalities
 6.2 Convergence in the rth Mean, Uniform Continuity, Uniform Integrability, and their Relationships
 Chapter 7. The Hahn–Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon–Nikodym Theorem
 Abstract
 7.1 The Hahn–Jordan Decomposition Theorem
 7.2 The Lebesgue Decomposition Theorem
 7.3 The Radon–Nikodym Theorem
 Chapter 8. Distribution Functions and Their Basic Properties, Helly–Bray Type Results
 Abstract
 8.1 Basic Properties of Distribution Functions
 8.2 Weak Convergence and Compactness of a Sequence of Distribution Functions
 8.3 Helly–Bray Type Theorems for Distribution Functions
 Chapter 9. Conditional Expectation and Conditional Probability, and Related Properties and Results
 Abstract
 9.1 Definition of Conditional Expectation and Conditional Probability
 9.2 Some Basic Theorems About Conditional Expectations and Conditional Probabilities
 9.3 Convergence Theorems and Inequalities for Conditional Expectations
 9.4 Further Properties of Conditional Expectations and Conditional Probabilities
 Chapter 10. Independence
 Abstract
 10.1 Independence of Events, σFields, and Random Variables
 10.2 Some Auxiliary Results
 10.3 Proof of Theorem 1 and of Lemma 1 in Chapter 9
 Chapter 11. Topics from the Theory of Characteristic Functions
 Abstract
 11.1 Definition of the Characteristic Function of a Distribution and Basic Properties
 11.2 The Inversion Formula
 11.3 Convergence in Distribution and Convergence of Characteristic Functions—The Paul Lévy Continuity Theorem
 11.4 Convergence in Distribution in the Multidimensional Case—The Cramér–Wold Device
 11.5 Convolution of Distribution Functions and Related Results
 11.6 Some Further Properties of Characteristic Functions
 11.7 Applications to the Weak Law of Large Numbers and the Central Limit Theorem
 11.8 The Moments of a Random Variable Determine its Distribution
 11.9 Some Basic Concepts and Results from Complex Analysis Employed in the Proof of Theorem 11
 Chapter 12. The Central Limit Problem: The Centered Case
 Abstract
 12.1 Convergence to the Normal Law (Central Limit Theorem, CLT)
 12.2 Limiting Laws of L(Sn) Under Conditions (C)
 12.3 Conditions for the Central Limit Theorem to Hold
 12.4 Proof of Results in Section 12.2
 Chapter 13. The Central Limit Problem: The Noncentered Case
 Abstract
 13.1 Notation and Preliminary Discussion
 13.2 Limiting Laws of L(Sn) Under Conditions (C″)
 13.3 Two Special Cases of the Limiting Laws of L(Sn)
 Chapter 14. Topics from Sequences of Independent Random Variables
 Abstract
 14.1 Kolmogorov Inequalities
 14.2 More Important Results Toward Proving the Strong Law of Large Numbers
 14.3 Statement and Proof of the Strong Law of Large Numbers
 14.4 A Version of the Strong Law of Large Numbers for Random Variables with Infinite Expectation
 14.5 Some Further Results on Sequences of Independent Random Variables
 Chapter 15. Topics from Ergodic Theory
 Abstract
 15.1 Stochastic Process, the Coordinate Process, Stationary Process, and Related Results
 15.2 MeasurePreserving Transformations, the Shift Transformation, and Related Results
 15.3 Invariant and Almost Sure Invariant Sets Relative to a Transformation, and Related Results
 15.4 MeasurePreserving Ergodic Transformations, Invariant Random Variables Relative to a Transformation, and Related Results
 15.5 The Ergodic Theorem, Preliminary Results
 15.6 Invariant Sets and Random Variables Relative to a Process, Formulation of the Ergodic Theorem in Terms of Stationary Processes, Ergodic Processes
 Chapter 16. Two Cases of Statistical Inference: Estimation of a RealValued Parameter, Nonparametric Estimation of a Probability Density Function
 Abstract
 16.1 Construction of an Estimate of a RealValued Parameter
 16.2 Construction of a Strongly Consistent Estimate of a RealValued Parameter
 16.3 Some Preliminary Results
 16.4 Asymptotic Normality of the Strongly Consistent Estimate
 16.5 Nonparametric Estimation of a Probability Density Function
 16.6 Proof of Theorems 3–5
 Appendix A. Brief Review of Chapters 1–16
 Chapter 1 Certain Classes of Sets, Measurability, and Pointwise Approximation
 Chapter 2 Definition and Construction of a Measure and its Basic Properties
 Chapter 3 Some Modes of Convergence of Sequences of Random Variables and their Relationships
 Chapter 4 The Integral of a Random Variable and its Basic Properties
 Chapter 5 Standard Convergence Theorems, The Fubini Theorem
 Chapter 6 Standard Moment and Probability Inequalities, Convergence in the rth Mean and its Implications
 Chapter 7 The Hahn–Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon–Nikodym Theorem
 Chapter 8 Distribution Functions and Their Basic Properties, Helly–Bray Type Results
 Chapter 9 Conditional Expectation and Conditional Probability, and Related Properties and Results
 Chapter 10 Independence
 Chapter 11 Topics from the Theory of Characteristic Functions
 Chapter 12 The Central Limit Problem: The Centered Case
 Chapter 13 The Central Limit Problem: The Noncentered Case
 Chapter 14 Topics from Sequences of Independent Random Variables
 Chapter 15 Topics from Ergodic Theory
 Chapter 16 Two Cases of Statistical Inference: Estimation of a Realvalued Parameter, Nonparametric Estimation of a Probability Density Function
 Appendix B. Brief Review of Riemann–Stieltjes Integral
 Appendix C. Notation and Abbreviations
 Selected References
 Revised Answers Manual to an Introduction to MeasureTheoretic Probability
 Chapter 1
 Chapter 2
 Chapter 3
 Chapter 4
 Chapter 5
 Chapter 6
 Chapter 7
 Chapter 8
 Chapter 9
 Chapter 10
 Chapter 11
 Chapter 12
 Chapter 13
 Chapter 14
 Chapter 15
 Index
Details
 No. of pages:
 426
 Language:
 English
 Copyright:
 © Academic Press 2014
 Published:
 24th March 2014
 Imprint:
 Academic Press
 eBook ISBN:
 9780128002902
 Hardcover ISBN:
 9780128000427
About the Author
George Roussas
George G. Roussas earned a B.S. in Mathematics with honors from the University of Athens, Greece, and a Ph.D. in Statistics from the University of California, Berkeley. As of July 2014, he is a Distinguished Professor Emeritus of Statistics at the University of California, Davis. Roussas is the author of five books, the author or coauthor of five special volumes, and the author or coauthor of dozens of research articles published in leading journals and special volumes. He is a Fellow of the following professional societies: The American Statistical Association (ASA), the Institute of Mathematical Statistics (IMS), The Royal Statistical Society (RSS), the American Association for the Advancement of Science (AAAS), and an Elected Member of the International Statistical Institute (ISI); also, he is a Corresponding Member of the Academy of Athens. Roussas was an associate editor of four journals since their inception, and is now a member of the Editorial Board of the journal Statistical Inference for Stochastic Processes. Throughout his career, Roussas served as Dean, Vice President for Academic Affairs, and Chancellor at two universities; also, he served as an Associate Dean at UCDavis, helping to transform that institution's statistical unit into one of national and international renown. Roussas has been honored with a Festschrift, and he has given featured interviews for the Statistical Science and the Statistical Periscope. He has contributed an obituary to the IMS Bulletin for ProfessorAcademician David Blackwell of UCBerkeley, and has been the coordinating editor of an extensive article of contributions for Professor Blackwell, which was published in the Notices of the American Mathematical Society and the Celebratio Mathematica.
Affiliations and Expertise
University of California, Davis, USA
Reviews
"...a very thorough discussion of many of the pillars of the subject, showing in particular how 'measure theory with total measure one' is just the tip of the iceberg...It’s quite a book."MAA.org, An Introduction to MeasureTheoretic Probability
"This second edition employs a classical approach to teaching students of statistics, mathematics, engineering, econometrics, finance, and other disciplines measuretheoretic probability…requires no prior knowledge of measure theory, discusses 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."Zentralblatt MATH 12871
"...provides basic tools in measure theory and probability, in the classical spirit, relying heavily on characteristic functions as tools without using martingale or empirical process methods. A wellwritten book. Highly recommended [for] graduate students; faculty."CHOICE
"Based on the material presented in the manuscript, I would without any hesitation adopt the published version of the book. The topics dealt are essential to the understanding of more advanced material; the discussion is deep and it is combined with the use of essential technical details. It will be an extremely useful book. In addition it will be a very popular book."Madan Puri, Indiana University
"Would likely use as one of two required references when I teach either Stat 709 or Stat 732 again. Would also highly recommend to colleagues. The author has written other excellent graduate texts in mathematical statistics and contiguity and this promises to be another. This book could well become an important reference for mathematical statisticians."Richard Johnson, University of Wisconsin
"The author has succeeded in making certain deep and fundamental ideas of probability and measure theory accessible to statistics majors heading in the direction of graduate studies in statistical theory."Doraiswamy Ramachandran, California State University