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New York: Chapman and Hall/CRC, 1994. — 632p
Volume I of this two-volume text and reference work begins by providing a foundation in measure and integration theory. It then offers a systematic introduction to probability theory, and in particular, those parts that are used in statistics. This volume discusses the law of large numbers for independent and non-independent random variables, transforms, special distributions, convergence in law, the central limit theorem for normal and infinitely divisible laws, conditional expectations and martingales. Unusual topics include the uniqueness and convergence theorem for general transforms with characteristic functions, Laplace transforms, moment transforms and generating functions as special examples. The text contains substantive applications, e.g., epidemic models, the ballot problem, stock market models and water reservoir models, and discussion of the historical background. The exercise sets contain a variety of problems ranging from simple exercises to extensions of the theory.
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Chapman& Hal l Pr obabi l i t y Se r i e s
PROBABI LI TY WI TH AVI EW TOWARD STATI STI CS VOLUMEI
J .Hof f mannJ or g e ns e n
PROBABILITY WITH A VIEW TOWARD STATISTICS VOLUME I
Chapman & Hall Probability Series Series Editors
Jean Bertoin, Universite Paris VI David S. Griffeath, University of Wisconsin Marjorie G. Hahn, Tufts University J. Michael Steele, University of Pennsylvania Hoffmann-Jergensen, l, Probability with a View toward Statistics,
Volumes I and II Stromberg, K., Probability for Analysts
PROBABILITY WITH A VIEW TOWARD STATISTICS VOLUME I
J. Hoffmann-J0rgensen University of Aarhus
Springer-Science+Business Media, B.Y.
Copyright © 1994 by Springer Science+Business Media Dordrecht Originally published by Chapman & Hall in 1994. Softcover reprint of the hardcover 1st edition 1994 All rights reserved. No part of this book may be reprinted or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or by an information storage or retrieval system, without permission in writing from the publishers. Library of Congress Cataloging in Publication Data Hoffmann-J\?lrgensen, J. (J\?lrgen) ,1929Probability with a view toward statistics / 1. Hoffmann-J\?lrgensen p. cm. Includes index. ISBN 978-0-412-05221-7 ISBN 978-1-4899-3019-4 (eBook) DOI 10.1007/978-1-4899-3019-4
1. Probabilities. I. Title. QA273.H689 1994 519.2-dc20
2. Mathematical statistics.
94-18943 CIP
British Library Cataloguing in Publication Data available
Contents Preface . ..
. xiii
Introduction .
. xvi
References to the History of Probability .
xxxix
Chapter 1,' MEASURE THEORY . ...
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1,33 1.34 1.35 1.36
1 3
Pavings and Set Functions . Stability of Pavings . . . Contents . . . . . . . . . . . Measures . . . . . . . . . . . Generation of Algebras and IT-Algebras Sierpinsky's Stability Lemma . The Uniqueness Theorem I . . . . . . . . . The Borel IT-Algebra . . . . . . . . . . . . Generation of the Borel IT-Algebra ... . The Borel IT-Algebra on Rand Rk .. . Finite IT-Algebras . . . . . . . . . . . . . . The IT-Algebra Generated by Singletons . The Counting Measure . Measures on (N,2N) .... The Dirac Measure . . . . . . Construction of Algebras. . . The Borel IT-Algebra Revisited Construction of IT-Algebras .. Null Sets and Almost Sure Properties Operations on Measures Set Functions . . . . . . . The p,-Measurable Sets .. Caratheodory's Theorem. Additivity