Hilbert and Banach Space-Valued Stochastic Processes

Description

This is a development of the book entitled Multidimensional Second Order Stochastic Processes. It provides a research expository treatment of infinite-dimensional stationary and nonstationary stochastic processes or time series, based on Hilbert and Banach space-valued second order random variables. Stochastic measures and scalar or operator bimeasures are fully discussed to develop integral representations of various classes of nonstationary processes such as harmonizable, V-bounded, Cramér and Karhunen classes as well as the stationary class. A new type of the Radon–Nikodým derivative of a Banach space-valued measure is introduced, together with Schauder basic measures, to study uniformly bounded linearly stationary processes. Emphasis is on the use of functional analysis and harmonic analysis as well as probability theory. Applications are made from the probabilistic and statistical points of view to prediction problems, Kalman filter, sampling theorems and strong laws of large numbers. Generalizations are made to consider Banach space-valued stochastic processes to include processes of pth order for p ≥ 1. Readers may find that the covariance kernel is always emphasized and reveals another aspect of stochastic processes. This book is intended not only for probabilists and statisticians, but also for functional analysts and communication engineers. CONTENTS Preface I. Introduction and preliminaries 1.1. Stationary processes 1.2. Harmonizable processes 1.3. Multidimensional and other extensions Bibliographical notes II. Hilbert modules and covariance kernels 2.1. Normal Hilbert B(H)-modules 2.2. Submodules, operators and functionals 2.3. Characterization and structure 2.4. Positive definite kernels and reproducing kernel spaces 2.5. Harmonic analysis for normal Hilbert B(H)-modules Bibliographical notes III. Stochastic measures and operator-valued bimeasures 3.1. Semivariations and variations 3.2. Orthogonally scattered dilations 3.3. Gramian orthogonally scattered dilations 3.4. The spaces L1(F) and L2(F) 3.5. L2-spaces for bimeasures 3.6. Riesz type theorems 3.7. Weak topology on measures 3.8. A Chouquet type theorem Bibliographical notes IV. Radon-Nikodým derivatives and Schauder basic measures 4.1. Pseudo Radon-Nikodým derivatives (1) 4.2. Pseudo Radon-Nikodým derivatives (2) 4.3. Schauder basic measures 4.4. Gramian Schauder basic measures Bibliographical notes V. Multidimensional stochastic processes 5.1. General concepts 5.2. Stationary processes 5.3. Harmonizable processes 5.4. V-bounded processes 5.5. Cramér and Karhunen classes 5.6. Operator representations 5.7. Series representations 5.8. Moving average representations 5.10. Subordination Bibliographical notes VI. Special topics 6.1. Wold decompositions 6.2. Cramér decompositions 6.3. The KF-class 6.4. Gramian uniformly bounded linearly stationary processes 6.5. Periodically correlated processes 6.6. Absolutely summing processes 6.7. Final remarks Bibliographical notes VII. Applications 7.1. Prediction problems 7.2. Kalman filter 7.3. Sampling theorems 7.4. Strong laws of large numbers Bibliographical notes VIII. Generalizations 8.1. Banach space-valued random variables 8.2. The spaces B(U, H) and B(U, U*) 8.3. B(U, H)-valued measures 8.4. B(U, U*)-valued measures and bimeasures 8.5. B(U, H)-valued processes 8.6. B(U, V)-valued measures 8.7. B(U, V)-valued processes Bibliographical notes References Indices Notation Index Author Index Subject Index