Limit Theorems for Stochastic Processes
Initially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is on
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Editors
M. Artin S. S. Chern J. M. Fröhlich E. Heinz H. Hironaka F. Hirze bruch L. Hörmander S. MacLane C. C. Moore J. K. Moser M. Nagata W. Schmidt D.S. Scott Ya.G. Sinai J. Tits M. Waldschmidt S. Watanabe Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Limit theorems ... (by courtesy ofProfessor A. T. Fomenko ofthe Moscow State University)
Jean Jacod Albert N. Shiryaev
Linaitllheorenas for Stochastic Processes
Springer-Verlag Berlin Heidelberg GmbH
Jean Jacod Laboratoire de ProbabiliU:s Tours 56 (3e etage) 4, Place Jussieu F-75230 Paris 05, France Albert N. Shiryaev Steklov Mathematical Institute Vavilova42 GSP-l, 117966 Moscow, USSR
Mathematics Subject Classification (1980): 60-F-17
ISBN 978-3-662-02516-1
Library of Congress Cataloging-in-Publication Data Jacod, Jean. Limit theorems for stochastic processes. (Grundlehren der mathematischen Wissenschaften ; 288) Bibliography: p. lncludes indexes. 1. Semimartingales (Mathematics) 2. Limit theorems (Probability theory) L Shirîaev, Al'bert Nikolaevich. II. Title. III. Series. QA274.5.J33 1987 519.2'87 87-9865 ISBN 978-3-662-02516-1 ISBN 978-3-662-02514-7 (eBook) DOI 10.1007/978-3-662-02514-7 This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concemed, specifically those of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication ar parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in ils version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act ofthe German Copyright Law. © Springer-Verlag Berlin Heidelberg 1987
Originally published by Springer-Verlag Berlin Heide1berg New York in 1987 Softcover reprint ofthe hardeaver lst edition 1987 Typesetting: Asea Trade Typesetting Ltd., Hong Kong 2141/3140-543210
To oursons 0/ivier:J Vincent and Andrei
Introduction
The limit theorems in this book belong to the theory of weak convergence of probability measures on metric spaces. More precisely, our main aim is to give a systematic exposition of the theory of convergence in law for those stochastic processes that are semimartingales. The choice of the class of semimartingales as our chief object of study has two reasons. One is that this class is broad enough to accomodate most common processes: discrete-time processes, diffusions, many Markov processes, point processes, solutions of stochastic differential equations, ... Our second reason is that we have in our hands a very powerful tool for studying these processes, namely the stochastic calculus. Since the theory of semimartingales, and related topics as random measures, are not usually associated with limit theorems, we decided to write a rather complete account of that theory, which is covered in the first two chapters. In particular, we devote much space to a careful and detailed exposition of the notion of characterist
