# Donsker theorems for diffusions: Necessary and sufficient conditions

@article{Vaart2005DonskerTF, title={Donsker theorems for diffusions: Necessary and sufficient conditions}, author={Aad van der Vaart and Harry van Zanten}, journal={Annals of Probability}, year={2005}, volume={33}, pages={1422-1451} }

We consider the empirical process 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 G t converge weakly to those of a zero-mean Gaussian random process G. We prove that the weak convergence G t ⇒ G takes place in l∞(F) if and only if the limit G exists as a tight, Borel measurable map. The proof relies on majorizing measure techniques for continuous…

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