# 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}
}```
• Published 1 July 2005
• Mathematics
• Annals of Probability
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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