A Comparison of Continuity Conditions for Gaussian Processes

Abstract

Three sufficient conditions for continuity of real-valued, separable, Gaussian processes on $\mathbb{R}^1$ are compared. They are: (1) Fernique's (1964) integral condition, (2) the Kahane (1960)-Nisio (1969) condition on the spectrum of stationary processes and (3) Dudley's (1967) condition involving $\varepsilon$-entropy. Let $S_1 \equiv$ set of stationary, separable, Gaussian processes that can be proven continuous by condition $i = 1, 2, 3$. Dudley (1967) has shown that $S_1 \subseteq S_3$. It is shown here that $S_2 \subset S_1 \subset S_3$, that is, the inclusion is strict. These results extend to non-stationary processes where appropriate. The Kahane-Nisio condition is strengthened and the best possible integral condition for continuity involving the spectrum is given. A result on the $\varepsilon$-entropy of blocks in a separable Hilbert space is also of independent interest.