Probability and Stochastics
This text is an introduction to the modern theory and applications of probability and stochastics. The style and coverage is geared towards the theory of stochastic processes, but with some attention to the applications. In many instances the gist of the
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Editorial Board S. Axler K.A. Ribet
For other titles in this series, go to http://www.springer.com/series/136
Erhan C¸ınlar
Probability and Stochastics
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Erhan C¸ınlar Princeton University 328 Sherrerd Hall Princeton, NJ 08544 USA [email protected] Editorial Board: S. Axler San Francisco State University Mathematics Department San Francisco, CA 94132 USA [email protected]
K. A. Ribet University of California at Berkeley Mathematics Department Berkeley, CA 94720 USA [email protected]
ISSN 0072-5285 ISBN 978-0-387-87858-4 e-ISBN 978-0-387-87859-1 DOI 10.1007/978-0-387-87859-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011921929 Mathematics Subject Classification (2010): 60 c Springer Science+Business Media, LLC 2011 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 excerpts 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 on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface This is an introduction to the modern theory of probability and stochastic processes. The aim is to enable the student to have access to the many excellent research monographs in the literature. It might be regarded as an updated version of the textbooks by Breiman, Chung, and Neveu, just to name three. The book is based on the lecture notes for a two-semester course which I have offered for many years. The course is fairly popular and attracts graduate students in engineering, economics, physics, and mathematics, and a few overachieving undergraduates. Most of the students had familiarity with elementary probability, but it was safer to introduce each concept carefully and in a uniform style. As Martin Barlow put it once, mathematics attracts us because the need to memorize is minimal. So, only the more fundamental facts are labeled as theorems; they are worth memorizing. Most other results are put as propositions, comments, or exercises. Also put as exercises are results that can be understood only by doing the tedious work necessary. I believe in the Chinese proverb: I hear, I forget; I see, I remember; I do, I know. I have been considerate: I do not assume that the reader will go through the book line by line from the beginning to the end. Some things are recalled or re-introduced when they are needed. In each chapter or section, the essential material is put first, technical material is put toward the end. Subheadings
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