Kakihara - Hilbert And Banach Space-Valued Stochastic Processes: 13 (Series On Multivariate Analysis)

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Rao Vol. 13: Hilbert and Banach Space-Valued Stochastic Processes
Y. Kakihara

Title: Hilbert and Banach space-valued stochastic processes / Yuichiro Kakihara. Description: Hackensack, New Jersey : World Scientific, [2021] | Series: Series on multivariate analysis, 1793-1169 ; vol. 13 | Includes bibliographical references and index. Identifiers: LCCN 2020048414 | ISBN 9789811211744 (hardcover) | ISBN 9789811211751 (ebook for institutions) | ISBN 9789811211768 (ebook for individuals) Subjects: LCSH: Stochastic processes. | Hilbert space. | Functional analysis. | Functional analysis.

Classification: LCC QA274 .K325 2021 | DDC 519.2/3--dc23 LC record available at https://lccn.loc.gov/2020048414 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA.

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Univariate stationary stochastic processes in the weak sense were initiated by Khintchine in 1934, and since then the theory has been developed by many authors. In many applications, however, nonstationary processes play an important role. As an extension of stationarity Love, in the middle of the 1940s, introduced harmonizability and, in 1959, Rozanov gave another harmonizability, where Rao distinguished these two concepts by calling them strong and weak harmonizabilities in 1982. Another extension is to consider multidimensional stochastic processes. In the late 1950s Wiener and Masani developed such a theory for stationary stochastic processes. This book is concerned with infinite dimensional nonstationary (especially harmonizable) second order stochastic processes on (mainly) a locally compact abelian group.

Since harmonizable processes are the Fourier transforms of Hilbert space-valued measures, our treatment uses functional analysis as well as harmonic analysis as mentioned above. Here is a brief description of each chapter. In , normal Hilbert B(H)-modules are considered in detail, where H is a (separable) complex Hilbert space and B(H) is the algebra of all bounded linear operators on H. This kind of spaces is useful to describe infinite dimensional second order stochastic processes and is essentially the same as the spaces of Hilbert-Schmidt class operators between Hilbert spaces. In . With the preparation of deals with some special topics such as Wold and Cramr decompositions, and KF-processes.

Gramian uniformly bounded linearly stationary processes, periodically correlated processes and absolutely summing processes are of interest and are also treated in some detail. Some basic results on isotropic processes, processes on hypergroups, and processes on locally compact groups are stated without proof since a greater preparation is needed to give a full treatment of these topics. In , we explore the theory of Banach space-valued stochastic processes. We study second order stochastic processes and, in general, pth order ones. This is realized by dealing with operator-valued processes that the original processes induce. Then, we can obtain similar results as in the case of Hilbert space-valued stochastic processes.

This topic is not fully examined and is a promising field of study. Each chapter has the Bibliographical notes section assigning proper, and hopefully accurate, credits to authors whose contributions supported this book. A numbering system of this book should be mentioned. An item such as III.4.5 denotes the fifth in Section 4 of . In a given chapter, only the section and item numbers are used, and in a given section only the item number is used. M. M.

Rao of University of California at Riverside. The author would like to express his sincere gratitude to Professor Rao for giving him this great opportunity. Special thanks are due to Ms. Lai Fun Kwong and the Production Department of World Scientific Publishing Company for their help, cooperation and patience. Yichir KakiharaSan Bernardino, CaliforniaOctober, 2020