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Springer Texts in Statistics Advisors: George Casella Stephen Fienberg Ingram Olkin

Springer Texts in Statistics Alfred: Elements of Statistics for the Life and Social Sciences Athreya and Lahiri: Measure Theory and Probability Theory Berger: An Introduction to Probability and Stochastic Processes Bilodeau and Brenner: Theory of Multivariate Statistics Blom: Probability and Statistics: Theory and Applications Brockwell and Davis: Introduction to Times Series and Forecasting, Second Edition Carmona: Statistical Analysis of Financial Data in S-Plus Chow and Teicher: Probability Theory: Independence, Interchangeability, Martingales, Third Edition Christensen: Advanced Linear Modeling: Multivariate, Time Series, and Spatial Data—Nonparametric Regression and Response Surface Maximization, Second Edition Christensen: Log-Linear Models and Logistic Regression, Second Edition Christensen: Plane Answers to Complex Questions: The Theory of Linear Models, Third Edition Creighton: A First Course in Probability Models and Statistical Inference Davis: Statistical Methods for the Analysis of Repeated Measurements Dean and Voss: Design and Analysis of Experiments du Toit, Steyn, and Stumpf: Graphical Exploratory Data Analysis Durrett: Essentials of Stochastic Processes Edwards: Introduction to Graphical Modelling, Second Edition Finkelstein and Levin: Statistics for Lawyers Flury: A First Course in Multivariate Statistics Ghosh, Delampady and Samanta: An Introduction to Bayesian Analysis: Theory and Methods Gut: Probability: A Graduate Course Heiberger and Holland: Statistical Analysis and Data Display: An Intermediate Course with Examples in S-PLUS, R, and SAS Jobson: Applied Multivariate Data Analysis, Volume I: Regression and Experimental Design Jobson: Applied Multivariate Data Analysis, Volume II: Categorical and Multivariate Methods Kalbfleisch: Probability and Statistical Inference, Volume I: Probability, Second Edition Kalbfleisch: Probability and Statistical Inference, Volume II: Statistical Inference, Second Edition Karr: Probability Keyfitz: Applied Mathematical Demography, Second Edition Kiefer: Introduction to Statistical Inference Kokoska and Nevison: Statistical Tables and Formulae Kulkarni: Modeling, Analysis, Design, and Control of Stochastic Systems Lange: Applied Probability Lange: Optimization Lehmann: Elements of Large-Sample Theory (continued after index)

Krishna B. Athreya Soumendra N. Lahiri Measure Theory and Probability Theory

Krishna B. Athreya Department of Mathematics and Department of Statistics Iowa State University Ames, IA 50011 kba@iastate.edu Soumendra N. Lahiri Department of Statistics Iowa State University Ames, IA 50011 snlahiri@iastate.edu Editorial Board George Casella Department of Statistics University of Florida Gainesville, FL 32611-8545 Pittsburgh, PA 15213-3890 USA Stephen Fienberg Department of Statistics Carnegie Mellon University Stanford University Stanford, CA 94305 USA Ingram Olkin Department of Statistics USA Library of Congress Control Number: 2006922767 ISBN-10: 0-387-32903-X ISBN-13: 978-0387-32903-1 e-ISBN: 0-387-35434-4 Printed on acid-free paper. ©2006 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excepts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (MVY) 9 8 7 6 5 4 3 2 1 springer.com

Dedicated to our wives Krishna S. Athreya and Pubali Banerjee and to the memory of Uma Mani Athreya and Narayani Ammal

Preface This book arose out of two graduate courses that the authors have taught during the past several years; the ﬁrst one being on measure theory followed by the second one on advanced probability theory. The traditional approach to a ﬁrst course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the diﬀerentation theorems of Lebesgue, Lp-spaces on R, and do general mea- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case ﬁrst before going to the general one. But this also has the disadvantage in making many students’ perspective on mea- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of σ-algebra of sets and countable additivity of a set function are dictated by certain very natu- ral approximation procedures from practical applications and that they are not just some abstract ideas. Next, the general extension theorem of Carathedory is presented in Chapter 1. As immediate examples, the con- struction of the large class of Lebesgue-Stieltjes measures on the real line and Euclidean spaces is discussed, as are measures on ﬁnite and countable

viii Preface spaces. Concrete examples such as the classical Lebesgue measure and var- ious probability distributions on the real line are provided. This is further developed in Chapter 6 leading to the construction of measures on sequence spaces (i.e., sequences of random variables) via Kolmogorov’s consistency theorem. After providing a fairly comprehensive treatment of measure and inte- gration theory in the ﬁrst part (Introduction and Chapters 1–5), the focus moves onto probability theory in the second part (Chapters 6–13). The fea- ture that distinguishes probability theory from measure theory, namely, the notion of independence and dependence of random variables (i.e., measure- able functions) is carefully developed ﬁrst. Then the laws of large numbers are taken up. This is followed by convergence in distribution and the central limit theorems. Next the notion of conditional expectation and probability is developed, followed by discrete parameter martingales. Although the de- velopment of these topics is based on a rigorous measure theoretic founda- tion, the heuristic and intuitive backgrounds of the results are emphasized throughout. Along the way, some applications of the results from probabil- ity theory to proving classical results in analysis are given. These include, for example, the density of normal numbers on (0,1) and the Wierstrass approximation theorem. These are intended to emphasize the beneﬁts of studying both areas in a rigorous and combined fashion. The approach to conditional expectation is via the mean square approximation of the “unknown” given the “known” and then a careful approximation for the L1-case. This is a natural and intuitive approach and is preferred over the “black box” approach based on the Radon-Nikodym theorem. The ﬁnal part of the book provides a basic outline of a number of special topics. These include Markov chains including Markov chain Monte Carlo (MCMC), Poisson processes, Brownian motion, bootstrap theory, mixing processes, and branching processes. The ﬁrst two parts can be used for a two-semester sequence, and the last part could serve as a starting point for a seminar course on special topics. This book presents the basic material on measure and integration theory and probability theory in a self-contained and step-by-step manner. It is hoped that students will ﬁnd it accessible, informative, and useful and also that they will be motivated to master the details by carefully working out the text material as well as the large number of exercises. The authors hope that the presentation here is found to be clear and comprehensive without being intimidating. Here is a quick summary of the various chapters of the book. After giving an informal introduction to the ideas of measure and integration theory, the construction of measures starting with set functions on a small class of sets is taken up in Chapter 1 where the Caratheodory extension theorem is proved and then applied to construct Lebesgue-Stieltjes measures. Integra- tion theory is taken up in Chapter 2 where all the basic convergence the- orems including the MCT, Fatou, DCT, BCT, Egorov’s, and Scheﬀe’s are

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