Sample-Continuity of Square-Integrable Processes

Abstract

Let $\{X(t): t\in\lbrack 0, 1\rbrack\}$ be a stochastic process and $f$ a nonnegative function on [0, 1] which is nondecreasing in a neighborhood of 0. Under the assumption that $E(X(t) - X(s))^2 \leqq f(|t - s|)$, we find best possible conditions for determining whether or not $X(t)$ is sample-continuous.