An Introduction to Heavy-Tailed and Subexponential Distributions
Heavy-tailed probability distributions are an important component in the modeling of many stochastic systems. They are frequently used to accurately model inputs and outputs of computer and data networks and service facilities such as call centers. T
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Springer Series in Operations Research and Financial Engineering
Sergey Foss Dmitry Korshunov Stan Zachary
An Introduction to Heavy-Tailed and Subexponential Distributions
Springer Series in Operations Research and Financial Engineering Series Editors Thomas Mikosch Sidney I. Resnick Stephen M. Robinson
For other titles published in this series, go to http://www.springer.com/series/3182
Sergey Foss
•
Dmitry Korshunov
•
Stan Zachary
An Introduction to Heavy-Tailed and Subexponential Distributions
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Sergey Foss Department of Actuarial Mathematics Heriot-Watt University Riccarton, Edinburgh United Kingdom and Sobolev Institute of Mathematics Novosibirsk, Russia [email protected]
Stan Zachary Department of Actuarial Mathematics Heriot-Watt University Riccarton, Edinburgh United Kingdom [email protected]
Dmitry Korshunov Sobolev Institute of Mathematics of the Russian Academy of Sciences 4 Koptyuga pr. 630090 Novosibirsk Russia [email protected]
ISSN 1431-8598 ISBN 978-1-4419-9472-1 e-ISBN 978-1-4419-9473-8 DOI 10.1007/978-1-4419-9473-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011928798 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 text studies heavy-tailed distributions in probability theory, and especially convolutions of such distributions. The main goal is to provide a complete and comprehensive introduction to the theory of long-tailed and subexponential distributions which includes many novel elements and, in particular, is based on the regular use of the principle of a single big jump. Much of the material appears for the first time in text form, including: – The establishment of new relations between known classes of subexponential distributions and the introduction of important new classes – The development of some important new concepts, including those of h-insensitivity and local subexponentiality – The presentation of new and direct probabilistic proofs of known asymptotic results A number of recent textbooks and monographs contain some elements of the present theory, notably those by S. Asmussen [1, 2], P. Embrechts, C. Kl¨uppelberg, and T. Mikosch [22], T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels [42], an
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