Stationary random distributions

In the same way as the concept of distributions by L. Schwartz [1111 ) was introduced as an extended one of functions, we may define stationary random distributions as an extension of stationary random functions viz, stationary processes. Such consideration will enable u s to establish a unified theory of stationary processes, Brownian motion processes, processes with stationary increments and other similar ones, a s is shown in this p a p e r . We shall first introduce some fundamental notions in § 1. In § 2 we shall define the covariance distribution of stationary random distributions, which corresponds to Khintchine's covariance function [7]. I n § 3 a n d § 4 we shall prove the spectral decomposition theorems of covariance distributions and stationary random distributions respectively. In § 5 we shall discuss the derivatives of stationary distributions. In § 6 we shall show that any stationary distribution is identified with a k-th derivative ( in th e sense of distributions) of a certain continuous process with stationary k-th order increments for some k.