Stopped Random Walks Limit Theorems and Applications
Classical probability theory provides information about random walks after a fixed number of steps. For applications, however, it is more natural to consider random walks evaluated after a random number of steps. Stopped Random Walks: Limit Theorems and A
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Stopped Random Walks Limit Theorems and Applications
Second Edition
Springer Series in Operations Research and Financial Engineering Series Editors: Thomas V. Mikosch Sidney I. Resnick Stephen M. Robinson
For other titles published in this series, go to http://www.springer.com/series/3182
Allan Gut
Stopped Random Walks Limit Theorems and Applications Second Edition
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Allan Gut Department of Mathematics Uppsala University SE-751 06 Uppsala Sweden [email protected]
Series Editors: Thomas V. Mikosch University of Copenhagen Laboratory of Actuarial Mathematics DK-1017 Copenhagen Denmark [email protected]
Stephen M. Robinson University of Wisconsin-Madison Department of Industrial Engineering Madison, WI 53706 USA [email protected]
Sidney I. Resnick Cornell University School of Operations Research and Industrial Engineering Ithaca, NY 14853 USA [email protected]
ISSN 1431-8598 ISBN 978-0-387-87834-8 DOI 10.1007/978-0-387-87835-5
e-ISBN 978-0-387-87835-5
Library of Congress Control Number: 2008942432 Mathematics Subject Classification (2000): 60G50, 60K05, 60F05, 60F15, 60F17, 60G40, 60G42 c Springer Science+Business Media, LLC 1988, 2009 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.com
Preface to the 1st edition
My first encounter with renewal theory and its extensions was in 1967/68 when I took a course in probability theory and stochastic processes, where the then recent book Stochastic Processes by Professor N.U. Prabhu was one of the requirements. Later, my teacher, Professor Carl-Gustav Esseen, gave me some problems in this area for a possible thesis, the result of which was Gut (1974a). Over the years I have, on and off, continued research in this field. During this time it has become clear that many limit theorems can be obtained with the aid of limit theorems for random walks indexed by families of positive, integer valued random variables, typically by families of stopping times. During the spring semester of 1984 Professor Prabhu visited Uppsala and very soon got me started on a book focusing on this aspect. I wish to thank him for getting me into this project, for his advice and suggestions, as well as his kindness and hospitality during my stay at Cornell in the spring of 1985. Throughout the writing of this book I have had immense help and support from Svante Janson. He
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