Ph . D . dissertation under the direction of Stamatis

Consider a stochastic process {x(t), t€T} of random elements of a Hilbert space H, whose index set is a locally compact Hausdorff space. The results obtained in this work fall into two broad categories, first the study of weakly stationary processes and their representations, and secondly the study of the sample path properties of not necessarily . stationary processes. In each case, we choose the index set T and the Hilbert spa~e H to be spaces appropriate to the investigation in hand. We first deal with stationary complex valued stochastic processes indexed by a locally compact topological group G, and consider their spectral representations. We show how the representation may be "inverted" to obtain an expression for the spectral measure; this result may be regarded as a weak law of large numbers. We give a decomposition of an arbitrary weakly stationary process into mean square continuous (m.s.c.) and "almost" uncorrelated components, and also show how the covariance function of a m.s.c. weakly stationary process may be estimated from the sample values via a strong law of large numbers. We then turn to a discussion of path properties, and show that any measurable complex valued process indexed by an arbitrary locally compact Hausdorff space may be approximated as closely as we please in a certain sense by a measurable process with continuous paths. This result is the analog of the well known measure theoretic fact that any measurable function i~r "almost" 'continuous (see e. g. Royden (1963) p. 57). Next we return to the group-indexed stationary processes, and find conditions on their spectral measures that are sufficient for path continuity with probability one. A special case of such processes are the so-called band limited processes we show that group indexed band 1imi~ed processes admit a sampling expansion, as do band limited processes over the real line. 14/e a.lso discuss band limited processes that are not necessarily stationary, extending an idea of .Zakai (1965). We characterize such non-stationary band limited processes and show they admit a modified sampling expansion. Finally we deal with the concept of group indexed second order processes having values in a separable Hilbert space. We prove a general representation theorem for such processes and from it derive a spectral representation for stationary processes. We prove a Hilbert space analog of the decomposition theorem mentioned above and lastly extend some of the results on path properties to the case of Hilbert space valued processes. •