AN APPROXIMATE MOVING AVERAGE REPRESENTATION OF THE PERIODIC STOCHASTIC PROCESS

Abstract

This paper presents a moving average of independent random variables with normal distributions that approximates a stochastic process whose sample paths are periodic (we call it the periodic stochastic process). Since the periodic stochastic process does not have a spectral density, it can not be directly represented as a moving average according to the Wold decomposition theorem. The results of this paper are twofold. First, we point out that the theorem originally proved by Slutzky (1937) is not satisfactory in the sense that the moving average process constructed by him does not converge to any processes in L 2 as the sum of white noise goes to infinity though the spectral distribution of it weakly converges to a step function which is the spectral distribution of a periodic stochastic process. Secondly we propose a new moving average process that approximates a nontrivial periodic stochastic process in L 2 and almost surely.