Donsker theorems for diffusions: Necessary and sufficient conditions

Abstract

We consider the empirical process $\mathbb{G}_{t}$ of a one-dimensional diffusion with finite speed measure, indexed by a collection of functions ℱ. By the central limit theorem for diffusions, the finite-dimensional distributions of $\mathbb{G}_{t}$ converge weakly to those of a zero-mean Gaussian random process $\mathbb{G}$. We prove that the weak convergence $\mathbb{G}_{t}\Rightarrow \mathbb{G}$ takes place in ℓ^∞(ℱ) if and only if the limit $\mathbb{G}$ exists as a tight, Borel measurable map. The proof relies on majorizing measure techniques for continuous martingales. Applications include the weak convergence of the local time density estimator and the empirical distribution function on the full state space.