Superconcentration and Related Topics
A certain curious feature of random objects, introduced by the author as “super concentration,” and two related topics, “chaos” and “multiple valleys,” are highlighted in this book. Although super concentration has established itself as a recognized featu
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Sourav Chatterjee
Superconcentration and Related Topics
Springer Monographs in Mathematics
For further volumes: www.springer.com/series/3733
Sourav Chatterjee
Superconcentration and Related Topics
Sourav Chatterjee Department of Statistics Stanford University Stanford, CA, USA
ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-319-03885-8 ISBN 978-3-319-03886-5 (eBook) DOI 10.1007/978-3-319-03886-5 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014930163 Mathematics Subject Classification (2010): 60E15, 60K35, 60G15, 82B44, 60G60, 60G70 © Springer International Publishing Switzerland 2014 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)
Preface
Understanding the fluctuations of random objects is one of the major goals of probability theory. There is a whole subfield of probability and analysis, called concentration of measure, devoted to understanding fluctuations of random objects. Measure concentration has seen tremendous progress in the last forty years. And yet, there is a large class of problems in which classical concentration of measure gives suboptimal bounds on the order of fluctuations. In 2008 and 2009, I posted two preprints on arXiv where it was shown that the sub
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