Probability Theory: Independence, Interchangeability, Martingales

✍ Scribed by Yuan Shih Chow, Henry Teicher (auth.)

• Publisher: Springer-Verlag New York
• Year: 1997
• Tongue: English
• Leaves: 504
• Series: Springer Texts in Statistics
• Edition: 3
• Category: Library

✦ Synopsis

Now available in paperback. This is a text comprising the major theorems of probability theory and the measure theoretical foundations of the subject. The main topics treated are independence, interchangeability,and martingales; particular emphasis is placed upon stopping times, both as tools in proving theorems and as objects of interest themselves. No prior knowledge of measure theory is assumed and a unique feature of the book is the combined presentation of measure and probability. It is easily adapted for graduate students familar with measure theory as indicated by the guidelines in the preface. Special features include: A comprehensive treatment of the law of the iterated logarithm; the Marcinklewicz-Zygmund inequality, its extension to martingales and applications thereof; development and applications of the second moment analogue of Wald's equation; limit theorems for martingale arrays, the central limit theorem for the interchangeable and martingale cases, moment convergence in the central limit theorem; complete discussion, including central limit theorem, of the random casting of r balls into n cells; recent martingale inequalities; Cram r-L vy theore and factor-closed families of distributions. This edition includes a section dealing with U-statistic, adds additional theorems and examples, and includes simpler versions of some proofs.

✦ Table of Contents

Front Matter....Pages i-xxii
Classes of Sets, Measures, and Probability Spaces....Pages 1-29
Binomial Random Variables....Pages 30-53
Independence....Pages 54-83
Integration in a Probability Space....Pages 84-112
Sums of Independent Random Variables....Pages 113-164
Measure Extensions, Lebesgue—Stieltjes Measure, Kolmogorov Consistency Theorem....Pages 165-209
Conditional Expectation, Conditional Independence, Introduction to Martingales....Pages 210-269
Distribution Functions and Characteristic Functions....Pages 270-312
Central Limit Theorems....Pages 313-353
Limit Theorems for Independent Random Variables....Pages 354-403
Martingales....Pages 404-443
Infinitely Divisible Laws....Pages 444-477
Back Matter....Pages 479-489

✦ Subjects

Probability Theory and Stochastic Processes