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July 2005 Donsker theorems for diffusions: Necessary and sufficient conditions
Aad van der Vaart, Harry van Zanten
Ann. Probab. 33(4): 1422-1451 (July 2005). DOI: 10.1214/009117905000000152


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.


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Aad van der Vaart. Harry van Zanten. "Donsker theorems for diffusions: Necessary and sufficient conditions." Ann. Probab. 33 (4) 1422 - 1451, July 2005.


Published: July 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1084.60047
MathSciNet: MR2150194
Digital Object Identifier: 10.1214/009117905000000152

Primary: 60F17 , 60J55 , 60J60 , 62M05

Keywords: continuous martingales , Diffusions , Donsker class , Local time , local time estimator , majorizing measures , uniform central limit theorem

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • July 2005
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