Fourier analysis of nonstationary stochastic processes

TATSUO KAWATA 1. Introduction.In the present paper we shall deal with the Fourier integral theory of the general stochastic processes.A general theory of covariance functions of a nonstationary process was developed by M. Love 8, 9 and some detailed discussions were also given by him on a special process called the harmoniable process, in which he covered the harmonic representation theorem.A number of papers dealing with nonstationary processes appeared in connection with noise theory or the statistical theory of regression (see E. Parzen 10).H. Cramer 4 extended the criterion concerning purely indeterministic stationary processes to the case of harmonisable processes and also gave a decomposition theorem.Still, it seems to me that very few papers, except those on the very general mathematical structure like measurability, separability, etc., were devoted to the study of general nonstationary processes.The aim of this paper is to develop the Fourier integral theory of the general nonstationary process X(x, co).Usually x is taken as a parameter which represents time, but in this paper we do not have to emphasize the time character of the process, cu represents an element of the given probability field fi.In fact, we shall prove some theorems which are analogous to those in the ordinary Ly theory of Fourier transform, such as summability theorems, inversion formulas for Fourier transforms and the Wiener formula.It will be of interest to notice that these theorems have some meaning from a probabilistic point of view.For instance, the summability theorems concerning stochastic processes will be thought of as the smoothing or filtering of these processes ; the Wiener formula will turn out to be some type of law of large numbers and the inversion formula for Fourier transforms will give the harmonic representation for the process.We shall suppose throughout this paper that the stochastic process X(x,co), -oo<x<oo, coeQ, being a probability field, satisfies the conditions : (i) it is measurable and separable, (ii) | X(x, co) |2 < oo for every x, (iii) ba E | X(x, co) \2dx< co for every finite interval (a, b), (iv) X(x,co) = Ofor every x and