An Introduction to Measure-theoretic ProbabilityBy
- George Roussas
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. The approach is classical, avoiding the use of mathematical tools not necessary for carrying out the discussions. All proofs are presented in full detail.
Graduate students primarily in statistics, mathematics, electrical & computer engineering or other information sciences; mathematical economics/finance in departments of economics.
Hardbound, 462 Pages
Published: October 2004
Imprint: Academic Press
"...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
- Preface1. 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 Case 13. The Central Limit Problem: The Noncentered Case14. Topics from Sequences of Independent Random Variables15. Topics from Ergodic Theory