Stationary hyperfunctional random process

Abstract

The theory of a generalized random process, i.e. random process in distributions was introduced by Gel’fand and Vilenkin. They made use of the Bochner–Schwartz type theorems for (conditionally) positive distributions, Schwartz kernel theorem and theorems on translation-invariant positive-definite bilinear functionals. We introduced a generalized random process in (Fourier) hyperfunctions by making use of our parallel results of the Bochner–Schwartz type theorems for (conditionally) positive Fourier hyperfunctions, Schwartz kernel theorem, and theorems on translation-invariant positive-definite bilinear functionals in Fourier hyperfunctions. We have an important guiding theme to compare the generalized random process in distributions in and our results in the generalized random process in (Fourier) hyperfunctions as follows; 1. the measures appearing in the generalized random process in distributions are tempered or polynomially increasing. 2. the measures appearing in the generalized random process in (Fourier) hyperfunctions are of infraexponential growth.