Probability Theory and Stochastic Processes
The ultimate objective of this book is to present a panoramic view of the main stochastic processes which have an impact on applications, with complete proofs and exercises. Random processes play a central role in the applied sciences, including operation
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Pierre Brémaud
Probability Theory and Stochastic Processes
Universitext
Universitext
Series Editors Sheldon Axler San Francisco State University Carles Casacuberta Universitat de Barcelona John Greenlees University of Warwick, Coventry Angus MacIntyre Queen Mary University of London
Kenneth Ribet University of California, Berkeley Claude Sabbah École Polytechnique, CNRS, Université Paris-Saclay, Palaiseau Endre Süli University of Oxford Wojbor A. Woyczyński, Case Western Reserve University
Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext.
More information about this series at http://www.springer.com/series/223
Pierre Brémaud
Probability Theory and Stochastic Processes
Pierre Brémaud Département d’Informatique INRIA, École Normale Supérieure Paris CX 5, France
ISSN 2191-6675 (electronic) ISSN 0172-5939 Universitext ISBN 978-3-030-40182-5 ISBN 978-3-030-40183-2 (eBook) https://doi.org/10.1007/978-3-030-40183-2 © Springer Nature Switzerland AG 2020 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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Contents xv
Introduction
1
Part One: Probability Theory 1 Warming Up 1.1 Sample Space, Events and Probability . . .
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