A note on the central limit theorem for square-integrable processes

Abstract:

A method is given for constructing sample-continuous processes which do not satisfy the central limit theorem in $C[0,1]$. Let $\{ X(t):t \in [0,1]\}$ be a stochastic process. Using our method we characterize all possible nonnegative functions f for which the condition \[ E( X(t) - X(s) )^2 \leqslant f( | t - s | )\] alone is sufficient to imply that $X(t)$ satisfies the central limit theorem in $C[0,1]$.