
Accession Number : AD0711082
Title : SPECTRAL DECOMPOSITION OF STOCHASTIC PROCESSES WITH PARAMETER IN A HILBERT SPACE.
Descriptive Note : Doctoral thesis,
Corporate Author : ILLINOIS UNIV URBANA COORDINATED SCIENCE LAB
Personal Author(s) : Gardner,Melvin Frank
Report Date : AUG 1970
Pagination or Media Count : 55
Abstract : 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 exp(ist)Z(ds), where Z sub s, s epsilon R) is an L sub 2process with orthogonal increments. The above integral is defined as the L sub 2limit of a sequence of approximating Riemann sums. The need arises to integrate with respect to some analogue of an L sub 2process 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.
Descriptors : (*STOCHASTIC PROCESSES, *MEASURE THEORY), HILBERT SPACE, GROUPS(MATHEMATICS), NUMERICAL INTEGRATION, CONVERGENCE, THEOREMS, THESES
Subject Categories : Statistics and Probability
Distribution Statement : APPROVED FOR PUBLIC RELEASE