Conditions for Sample-Continuity and the Central Limit Theorem

Abstract

Let $\{X(t): t\in\lbrack 0, 1\rbrack\}$ be a stochastic process. For any function $f$ such that $E(X(t) - X(s))^2 \leqq f(|t - s|)$, a condition is found which implies that $X$ is sample-continuous and satisfies the central limit theorem in $C\lbrack 0, 1\rbrack$. Counterexamples are constructed to verify a conjecture of Garsia and Rodemich and to improve a result of Dudley.