Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces
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672 Robert L. Taylor
Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces
Springer-Verlag Berlin Heidelberg New York 1978
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
672 Robert L. Taylor
Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces
Springer-Verlag Berlin Heidelberg New York 1978
Author Robert L. Taylor Department of Mathematics and Computer Science University of South Carolina Columbia, SC 292G8/USA
Library of Congress Cataloging in Publication Data
Taylor, Robert Lee, 1943Stochastic convergence of weighted sums of random elements in linear spaces. (Lecture notes in mathematics ; 672) Bibliography: p , Includes index. l. Linear topological Spaces. 2. Stochastic processes. 3. Limit theorems (Probability theory) 4. Law of large numbers. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 672. QA3.L28 no. 672 [QA273.43] 5l0'.8s [5l5' . 73] 78-l3024
AMS Subject Classifications (1970): 60B05, 60F15, 60G99
ISBN 3-540-08929-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08929-2 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-Yerlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE Recent interest in representing stochastic processes as random variables in function spaces has inspired the study of the "convergence of weighted sums" for random elements.
The purpose of these
notes is to provide a unified presentation of the results which have been obtained to date concerning the stochastic convergence of weighted sums of random elements in (primarily) linear topological spaces, including some recent work of the author and some colleagues. These notes are somewhat selfcontained with only a background knowledge of basic probability theory being absolutely essential, Chapter I presents many of the essential definitions and results from mathematical analysis and topology that are needed, and the definitions and properties of random elements are developed in Chapter II.
The remainder of the notes is concerned with the sto-
chastic convergence results and possible applications. Probabilists, statisticians, and mathematicians who are involved in research concerning convergence theory, stochastic processes, ergodic theory, and related topics will find these notes very
In addition, the notes will be useful for introducing advanced graduate students and researchers to the area of random elements and stochastic convergence of weighted sums in function spaces.
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