Regularly Varying Functions
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
508 Eugene Seneta
Regularly Varying Functions
Springer-Verlag Berlin· Heidelberg· New York 1976
Author Eugene Seneta Department of Statistics The Australian National University
P.O.Box4 Canberra, AC.T. 2600/Australia
AMS Subject Classifications (1970): 26A12, 26A48, 60E05 ISBN 3-540-07618-2 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-07618-2 Springer-Verlag New York · Heidelberg · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin ·Heidelberg 1976
Printed in Germany Printing and binding: Beltz, Offsetdruck, Hemsbach/Bergstr.
PREFACE The main purpose of these notes is to present, under precisely stated assumptions, the basia real-variable theory of regularly varying functions, in self-contained manner. Thus they may be used by any reader wishing to acquire a user's knowledge of this valuable analytical tool, irrespective of his field of mathematical specialization. With these aims in mind, the author has endeavoured to keep proofs simple where possible; and exercises have been provided to show the scope of the theory as well as to yield practice in the use of the material presented. The author's own interest in the subject matter was stimulated by probabilistic applications. The fundamental role played by Karamata's theory of regularly varying functions in probability theory was already suggested by the book of Gnedenko and Kolmogorov. It subsequently came to be widely recognized among probabilists with the publication in 1966 of Volume 2 of Feller's An Introduction to ProbabiZity Theory and Its AppZiaations which contained elements of an exposition of the Karamata theory. Unfortunately, this presentation was (and remains in the newer edition) highly personal, with precise assumptions and conditions unclear. It thus proves difficult for the non-expert reader. On the other hand, the papers in which Karamata's theory has been progressively refined and extended since the original contributions in the early 1930's, are so little known to probabilists that there is a general impression of their non-existence. It is the author's modest hope that these notes will help to bridge these gaps, in a manner somewhat different from de Haan' s (19 70a). Apart from the presentation of the basic theory, the reader will discern an attempt by the author to provide a selection of less standard material e.g. §2.4, and the Appendix. It needs to be mentioned also that the references given pertain only to the material presented, and so cannot in any sense be regarded as complete. The bulk of these no
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