Donsker theorems for diffusions: Necessary and sufficient conditions

Details Donsker theorems for diffusions: Necessary and sufficient conditions Donsker theorems for diffusions: Necessary and sufficient conditions

2005-07-21

arxiv.org/abs/math/0507412v1

Annals of Probability 2005, Vol. 33, No. 4, 1422-1451

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000152 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000152

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\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.

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