Continuity of Gaussian Processes and Random Fourier Series

Abstract

This paper is mainly a survey of results on the problem of finding necessary and sufficient conditions for a Gaussian process to be continuous. The relationship between this problem and the same one for random Fourier series is explored. Some new results are presented that give continuity conditions for stationary Gaussian processes in terms of the spectrum of the process. Let $X(t)$ be a real-valued stationary Gaussian process; $EX(t) = 0, EX^2(t) = 1$. Define $F$ by the equation $EX(t + h)X(t) = \int^\infty \cos \lambda h dF(\lambda)$. Assume that $F(\lambda)$ is concave for $\lambda \geqq\lambda_0 > 0$ then $X(t)$ is continuous a.s. if and only if $$\int^\infty \frac{(1 - F(x))^{\frac{1}{2}}}{x(\log x)^{\frac{1}{2}}} dx < \infty.$$ A similar result holds for Fourier series with normal coefficients.