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1111559922__99778899881111221111774444__TTPP..iinndddd 11 1122//77//2211 1100::0088 AAMM SERIES ON MULTIVARIATE ANALYSIS Editor: M M Rao ISSN: 1793-1169 Published Vol. 1: Martingales and Stochastic Analysis J. Yeh Vol. 2: Multidimensional Second Order Stochastic Processes Y. Kakihara Vol. 3: Mathematical Methods in Sample Surveys H. G. Tucker Vol. 4: Abstract Methods in Information Theory Y. Kakihara Vol. 5: Topics in Circular Statistics S. R. Jammalamadaka and A. SenGupta Vol. 6: Linear Models: An Integrated Approach D. Sengupta and S. R. Jammalamadaka Vol. 7: Structural Aspects in the Theory of Probability: A Primer in Probabilities on Algebraic-Topological Structures H. Heyer Vol. 8: Structural Aspects in the Theory of Probability (Second Edition) H. Heyer Vol. 9: Random and Vector Measures M. M. Rao Vol. 10: Abstract Methods in Information Theory (Second Edition) Y. Kakihara Vol. 11: Linear Models and Regression with R: An Integrated Approach D. Sengupta and S. R. Jammalamadaka Vol. 12: Stochastic Processes: Harmonizable Theory M. M. Rao Vol. 13: Hilbert and Banach Space-Valued Stochastic Processes Y. Kakihara LLaaiiFFuunn -- 1111559922 -- HHiillbbeerrtt aanndd BBaannaacchh SSppaaccee--VVaalluueedd SSttoocchhaassttiicc PPrroocceesssseess..iinndddd 11 1177//1111//22002200 99::1188::3399 aamm 1111559922__99778899881111221111774444__TTPP..iinndddd 22 1122//77//2211 1100::0088 AAMM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Kakihara, YûichirÔ, author. Title: Hilbert and Banach space-valued stochastic processes / Yûichirô 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. | Banach spaces--Radon-Nikodym property. | 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. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11592#t=suppl Printed in Singapore LLaaiiFFuunn -- 1111559922 -- HHiillbbeerrtt aanndd BBaannaacchh SSppaaccee--VVaalluueedd SSttoocchhaassttiicc PPrroocceesssseess..iinndddd 22 1177//1111//22002200 99::1188::3399 aamm December14,2020 9:50 11592-HilbertandBanachSpace-ValuedStochasticProcesses ws-book-combined page1 To my family Keiko Noriko, James and Sophie Satoko, Salvador and Yuzuko B1948 Governing Asia TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk BB11994488__11--AAookkii..iinndddd 66 99//2222//22001144 44::2244::5577 PPMM December14,2020 9:50 11592-HilbertandBanachSpace-ValuedStochasticProcesses ws-book-combined page3 PREFACE In 1997, the original version of this book was published under the title “Multidi- mensional Second Order Stochastic Processes.” A few years ago the author started to revise and expand the contents of the original book by adding two new chapters andsomeotherdevelopmentsthathaveoccurredinthelasttwentyyears,sothatthe new title of the book is “Hilbert and Banach Space-Valued Stochastic Processes.” The basic idea of this book is to treat infinite dimensional second order and pth orderstochasticprocessesusingfunctionalanalysis,operatortheory,vectormeasure theory and harmonic analysis as well as probability theory, which is the same as for the original version. We continue to use normal Hilbert modules for Hilbert and Banach space-valued second order stochastic processes. Let us begin with a historical note. 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 Loève, in the middleofthe1940s,introducedharmonizabilityand,in1959,Rozanovgaveanother harmonizability, whereRaodistinguishedthesetwoconceptsbycallingthemstrong 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- valuedmeasures,ourtreatmentusesfunctionalanalysisaswellasharmonicanalysis as mentioned above. Here is a brief description of each chapter. In Chapter I, a historical survey of stationary and nonstationary processes as well as multidimensional and other extensions are given. In Chapter II, 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 Chapter III, we investigate stochastic measures and operator-valued bimeasures. Let L2(Ω;H) be the Hilbert space of all H-valued random variables on a 0 probability space (Ω,F,µ) which are µ-square integrable. This space is a typical vii December14,2020 9:50 11592-HilbertandBanachSpace-ValuedStochasticProcesses ws-book-combined page4 viii PREFACE exampleofanormalHilbertB(H)-moduleandanL2(Ω;H)-valuedmeasurecanbe 0 regarded as an infinite dimensional stochastic measure. Such a measure induces a bimeasure with values in a subspace of B(H) and, in a certain sense, the converse is also true. Chapter IV is newly added. We introduce pseudo Radon-Nikody´m derivatives for any Banach space-valued measure ξ to include integration of not only scalar-valued functions but also operator-valued functions with respect to ξ. Then we can consider a kind of L1-space of operator-valued integrable functions, which enables us to define Schauder basic measures. This is applied to characterize (gramian) uniformly bounded linearly stationary processes in Chapter VI. With the preparation of Chapters I through IV, we can give the fundamental theory of finite and infinite dimensional stochastic processes in Chapter V. After defining stationarity, harmonizability and other nonstationarities, we discuss their integral representations and interrelations. As basic topics we study operator representations, series representations, moving average representations, approximation and convergence, and subordination. Chapter VI deals with some special topics such as Wold and Cramér decompositions, and KF-processes. Gramian uniformly bounded linearly stationary processes, periodically correlated processes and abso- lutely 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 Chapter VII, applications are made from probabilistic and statistical points of view to prediction problems, Kalman filter, sampling theorems and the strong laws of large numbers. Finally in Chapter VIII, we explore the theory of Banach space-valued stochastic processes. We study second order stochastic processes and, ingeneral,pthorderones. Thisisrealizedbydealingwithoperator-valuedprocesses thattheoriginalprocessesinduce. Then,wecanobtainsimilarresultsasinthecase of Hilbert space-valued stochastic processes. This topic is not fully examined and is a promising field of study. Eachchapterhasthe“Bibliographicalnotes”sectionassigningproper,andhope- fully 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 denotesthefifthinSection4ofChapterIII.Inagivenchapter,onlythesectionand item numbers are used, and in a given section only the item number is used. This book (original and current versions) was suggested by Professor M.M.Rao ofUniversityofCaliforniaatRiverside. Theauthorwouldliketoexpresshissincere 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. Yuîchirô Kakihara San Bernardino, California October, 2020 December14,2020 9:50 11592-HilbertandBanachSpace-ValuedStochasticProcesses ws-book-combined page5 CONTENTS Preface vii I. Introduction and preliminaries 1 1.1. Stationary processes ................................................. 1 1.2. Harmonizable processes .............................................. 4 1.3. Multidimensional and other extensions ............................... 9 Bibliographical notes ...................................................... 16 II. Hilbert modules and covariance kernels 19 2.1. Normal Hilbert B(H)-modules ....................................... 19 2.2. Submodules, operators and functionals ............................... 24 2.3. Characterization and structure ....................................... 29 2.4. Positive definite kernels and reproducing kernel spaces ............... 35 2.5. Harmonic analysis for normal Hilbert B(H)-modules ................. 50 Bibliographical notes ...................................................... 56 III. Stochastic measures and operator-valued bimeasures 57 3.1. Semivariations and variations ........................................ 57 3.2. Orthogonally scattered dilations ..................................... 79 3.3. Gramian orthogonally scattered dilations ............................ 92 3.4. The spaces L1(F) and L2(F) ....................................... 115 3.5. L2-spaces for bimeasures ............................................ 128 3.6. Riesz type theorems ................................................. 139 3.7. Weak topology on measures ......................................... 148 3.8. A Chouquet type theorem ........................................... 162 Bibliographical notes ..................................................... 166 IV. Radon-Nikody´m derivatives and Schauder basic measures 169 4.1. Pseudo Radon-Nikody´m derivatives (1) .............................. 169 (1) Radon-Nikody´m property ........................................ 170 (2) Pseudo Radon-Nikody´m derivatives .............................. 173 (3) Existence and uniqueness ........................................ 176 (4) Dunford-Schwartz type integration ............................... 185 ix