On vector measures related to deterministic non-stationary stochastic processes

Abstract

In this paper we consider a Lebesgue decomposition theorem for bounded vector measures and its application to the linear prediction problem of non-stationary stochastic processes. In Section 1 we present a proof of a Lebesgue decomposition theorem for bounded weakly compact vector measures with values in a Banach space by using the integration technique of vector measures introduced by Thomas [14]. The result is essentially due to Rickart [11]. In Section 2 we consider so-called p-majorizable bounded vector measures (see Pietsch [9]). We point out a relationship between the Lebesgue decomposition of a bounded p-majorizable weakly compact vector measure and the Lebesgue decompositions of its p-majorants. Our result generalizes a result of Pietsch [9] concerning bounded atomic p-majorizable vector measures. As an application of the results concerning bounded p-majorizable weakly compact vector measures we consider in Section 3 the linear prediction problem of such non-stationary stochastic processes, which are Fourier transforms of bounded stochastic measures. We show that certain properties, e.g. the singularity, of the spectral measure imply that the corresponding stochastic process is deterministic. Our result generalizes an analogous result of Pop-Stojanovic [10] concerning spectral measures of continuous stationary stochastic processes.