The Central Limit Theorem for Stochastic Processes

If f = {ft∣ t ∈ T} is a centered, second-order stochastic process with bounded sample paths, it is then known that f satisfies the central limit theorem in the topology of uniform convergence if and only if the intrinsic metric ρ2 f(on T) induced by f is totally bounded and the normalized sums are eventually uniformly ρ2 f-equicontinuous. We show that a centered, second-order stochastic process satisfies the central limit theorem in the topology of uniform convergence if and only if it has bounded sample paths and there exists totally bounded pseudometric ρ on T so that the normalized sums are eventually uniformly ρ-equicontinuous.