Continuous Time Markov Chains
In Chap. 1 we considered Markov chains Xn with a discrete time index n = 0, 1, 2, … In this chapter we will extend the notion to a continuous time parameter t ≥ 0, a setting that is more convenient for some applications. In discrete time we formulated th
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Richard Durrett
Essentials of Stochastic Processes Third Edition
Springer Texts in Statistics Series Editors: Richard DeVeaux Stephen E. Fienberg Ingram Olkin
More information about this series at http://www.springer.com/series/417
Richard Durrett
Essentials of Stochastic Processes Third Edition
123
Richard Durrett Mathematics, Duke University Durham, North Carolina, USA
ISSN 1431-875X ISSN 2197-4136 (electronic) Springer Texts in Statistics ISBN 978-3-319-45613-3 ISBN 978-3-319-45614-0 (eBook) DOI 10.1007/978-3-319-45614-0 Library of Congress Control Number: 2016951697 © Springer International Publishing Switzerland 1999, 2012, 2016 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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 of 20
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