Description

An Introduction to Measure-Theoretic Probability, Second Edition, employs a classical approach to teaching students of statistics, mathematics, engineering, econometrics, finance, and other disciplines that measure theoretic probability. This book 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. There is a considerable bend toward the way probability is actually used in statistical research, finance, and other academic and nonacademic applied pursuits. 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.

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 non-mathematical 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

Details

No. of pages:
426
Language:
English
Copyright:
© 2014
Published:
Imprint:
Academic Press
eBook ISBN:
9780128002902
Print 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 co-author of five special volumes, and the author or co-author 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 UC-Davis, 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 Professor-Academician David Blackwell of UC-Berkeley, 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 Measure-Theoretic Probability 

"This second edition employs a classical approach to teaching students of statistics, mathematics, engineering, econometrics, finance, and other disciplines measure-theoretic 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 1287-1
"...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 well-written 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 fundam