SPECTRAL DECOMPOSITION OF STOCHASTIC PROCESSES WITH PARAMETER IN A HILBERT SPACE.
ILLINOIS UNIV URBANA COORDINATED SCIENCE LAB
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The goal of this theses is to prove a spectral decomposition theorem for stochastic processes X sub t, to epsilon H where H is a real separable Hilbert space. If H R, the set of real numbers, then under certain conditions, namely stationarity of the process and continuity of the covariance function, the process may be written in the form X sub the integral over R of expistZds, where Z sub s, s epsilon R is an L sub 2-process with orthogonal increments. The above integral is defined as the L sub 2-limit of a sequence of approximating Riemann sums. The need arises to integrate with respect to some analogue of an L sub 2-process with orthogonal increments over a more general space than R. For this reason, the concept of a measure with orthogonal values henceforth abbreviated MOV is introduced. The definition of an integral with respect to a MOV is presented, and some basic properties of this integral are developed. The concept of weak convergence is extended to MOVs and analogues are proven of some of the usual thoerems about weak convergence. The spectral decomposition theorem for random processes X sub t, t epsilon H are proven.
- Statistics and Probability