An Introduction to Measure-Theoretic Probability

ISBN-13: 9780128000427 / Angielski / Twarda / 2014 / 426 str.

George Roussas

Opis

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.

Recenzje

"...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 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

Spis treści

1. Certain Classes of Sets, Measurability, Pointwise Approximation2. Definition and Construction of a Measure and Its Basic Properties3. Some Modes of Convergence of a Sequence of Random Variables and Their Relationships4. The Integral of a Random Variable and Its Basic Properties5. Standard Convergence Theorems, The Fubini Theorem6. Standard Moment and Probability Inequalities, Convergence in the r-th Mean and Its Implications7. The Hahn-Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and The Radon-Nikcodym Theorem8. Distribution Functions and Their Basic Properties, Helly-Bray Type Results9. Conditional Expectation and Conditional Probability, and Related Properties and Results10. Independence11. Topics from the Theory of Characteristic Functions12. The Central Limit Problem: The Centered Case13. The Central Limit Problem: The Noncentered Case14. Topics from Sequences of Independent Random Variables15. Topics from Ergodic Theory