Essentials of Stochastic Processes.pdf

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Springer Texts in Statistics Series Editors: G. Casella S. Fienberg I. Olkin For further volumes: http://www.springer.com/series/417

Richard Durrett Essentials of Stochastic Processes Second Edition 123

Richard Durrett Duke University Department of Mathematics Box 90320 Durham North Carolina USA ISSN 1431-875X ISBN 978-1-4614-3614-0 DOI 10.1007/978-1-4614-3615-7 Springer New York Heidelberg Dordrecht London ISBN 978-1-4614-3615-7 (eBook) Library of Congress Control Number: 2012937472 © Springer Science+Business Media, LLC 1999, 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

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Preface Between the ﬁrst undergraduate course in probability and the ﬁrst graduate course that uses measure theory, there are a number of courses that teach Stochastic Processes to students with many different interests and with varying degrees of mathematical sophistication. To allow readers (and instructors) to choose their own level of detail, many of the proofs begin with a nonrigorous answer to the question “Why is this true?” followed by a Proof that ﬁlls in the missing details. As it is possible to drive a car without knowing about the working of the internal combustion engine, it is also possible to apply the theory of Markov chains without knowing the details of the proofs. It is my personal philosophy that probability theory was developed to solve problems, so most of our effort will be spent on analyzing examples. Readers who want to master the subject will have to do more than a few of the 20 dozen carefully chosen exercises. This book began as notes I typed in the spring of 1997 as I was teaching ORIE 361 at Cornell for the second time. In Spring 2009, the mathematics department there introduced its own version of this course, MATH 474. This started me on the task of preparing the second edition. The plan was to have this ﬁnished in Spring 2010 after the second time I taught the course, but when May rolled around completing the book lost out to getting ready to move to Durham after 25 years in Ithaca. In the Fall of 2011, I taught Duke’s version of the course, Math 216, to 20 undergrads and 12 graduate students and over the Christmas break the second edition was completed. The second edition differs substantially from the ﬁrst, though curiously the length and the number of problems has remained roughly constant. Throughout the book there are many new examples and problems, with solutions that use the TI-83 to eliminate the tedious details of solving linear equations by hand. My students tell me I should just use MATLAB and maybe I will for the next edition. The Markov chains chapter has been reorganized. The chapter on Poisson processes has moved up from third to second, and is now followed by a treatment of the closely related topic of renewal theory. Continuous time Markov chains remain fourth, with a new section on exit distributions and hitting times, and reduced coverage of queueing networks. Martingales, a difﬁcult subject for students at this vii

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