Almost periodically unitary stochastic processes

A continuous second order complex process {X(t),t[epsilon]} defined on a probability space ([Omega], , P) is called almost periodically unitary (APU) if there exists a strongly continuous one parameter group of unitary operators {U([tau]),[tau][epsilon]} for which the set relatively dense (has bounded gaps) for every [var epsilon] > 0. These processes include continuous stationary processes for which S([var epsilon],X,U) = , continuous periodically correlated processes for which S([var epsilon],X,U)[superset or implies] {jT,j[epsilon]} for some real T, and the L2-valued uniformly almost periodic functions for which U([tau]) = In this paper, we show that X(t) is APU if and only if X(t) = U(t)[P(t)] where P(t) is an L2-valued uniformly almost periodic function. Examples are given and basic properties motivated by the theory of uniformly almost periodic functions are provided. These processes are shown to be uniformly almost periodically correlated and hence almost periodically correlated in the sense of Gladyshev. We give representations for the processes based on the spectral theory for unitary groups and on the harmonic analysis of uniformly almost periodic functions. Finally, we give an analysis of the correlation functions in terms of the representation theory, and show that every APU process is a uniform limit of a sequence of strongly harmonizable processes.