Hilbert and Banach Space-Valued Stochastic Processes

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: Introduction and PreliminariesHilbert Modules and Covariance KernelsStochastic Measures and Operator-Valued BimeasuresRadon–Nikodým Derivatives and Schauder Basic MeasuresMultidimensional Stochastic ProcessesSpecial TopicsApplicationsGeneralizations
Readership: Graduate students in mathematics, probabilists, statisticians, functional analysts, communication engineers and physicists.Infinite Dimensional Stochastic Processes;Normal Hilbert Modules;Stationary Processes;Harmonizable Processes;V-bounded Processes;Stationary Dilation;Series Representations;Cramér and Karhunen Classes;Wold and Cramér Decompositions;Uniformly Bounded Linearly Stationary Processes;Periodically Correlated Processes;Kalman Filter;Sampling Theorems;Strong Laws of Large Numbers;Representing Measures;Scalar and Operator Bimeasures;Reproducing Kernels0Key Features:Functional analysis methods are used on stochastic processesThis is an enlarged second edition with the latest updatesStructural analysis of nonstationary and stationary processes are also includedThis book is in the intersection of probability theory and analysis