Donsker theorems for diffusions: Necessary and sufficient conditions
Abstract
We consider the empirical process G_t of a onedimensional diffusion with finite speed measure, indexed by a collection of functions F. By the central limit theorem for diffusions, the finitedimensional distributions of G_t converge weakly to those of a zeromean Gaussian random process G. We prove that the weak convergence G_t\Rightarrow G takes place in \ell^{\infty}(F) if and only if the limit 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 2005
 arXiv:
 arXiv:math/0507412
 Bibcode:
 2005math......7412V
 Keywords:

 Mathematics  Probability;
 60J60;
 60J55;
 60F17;
 62M05 (Primary)
 EPrint:
 Published at http://dx.doi.org/10.1214/009117905000000152 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)