Representation of Stooiastic Processes of Second Order and Linear Qperaticjjs

Series representations are obtained for the entire class of measurable, second order stochastic processes defined on any interval of the real line. They include as particular cases all earlier representations; they suggest a notion of "smoothness" that generalizes well known continuity notions; and they decompose the stochastic process into two orthogonal parts, the smooth part and a strongly discontinuous part. Also linear operations on measurable, second order processes are studied; it is shown that all "smooth" linear operations on a process, and in particular all linear operations on a "smooth" process, can be approximated arbitrarily closely by linear operations on the sample paths of the process. Department of Statistics, University of North CaroZina, ChapeZ Hill, North CaroZina 2'1514. This research was supported by the NationaZ Science Foundation under Grant GU-2059 and by the Air Force Office of Saientifia Research under Grant AFOSR-68-1415.