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Quantum probability and the theory of operator algebras are both concerned with the study of noncommutative dynamics. Focusing on stationary processes with discrete-time parameter, this book presents (without many prerequisites) some basic problems of interest to both fields, on topics including extensions and dilations of completely positive maps, Markov property and adaptedness, endomorphisms of operator algebras and the applications arising from the interplay of these themes. Much of the material is new, but many interesting questions are accessible even to the reader equipped only with basic knowledge of quantum probability and operator algebras.
Quantum probability and the theory of operator algebras are both concerned with the study of noncommutative dynamics. Focusing on stationary processes with discrete-time parameter, this book presents (without many prerequisites) some basic problems of interest to both fields, on topics including extensions and dilations of completely positive maps, Markov property and adaptedness, endomorphisms of operator algebras and the applications arising from the interplay of these themes. Much of the material is new, but many interesting questions are accessible even to the reader equipped only with basic knowledge of quantum probability and operator algebras.
Stationary stochastic processes have been very useful in analyzing time series appear in applications. However in many engineering application and economic studies there are number of important time series that are not stationary. Hence several authors have been studied non-stationary processes. Several classes of non-stationary processes such as: (1) Harmonizable processes; (2) Periodically Correlated (PC) processes; (3) Almost Periodically Correlated (PAC) processes; and (4) Correlation Autoregressive (CAR) processes have been introduced and studied. In this project we have studying the non-stationary processes in general and the PC, APC, and CAR processes in particular obtaining several results which either have been published or submitted for publication in refereed journals.