2005/07/01 by Aad van der Vaart, Harry van Zanten
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #Markov Chains and Monte Carlo Methods #Stochastic processes and financial applications #math.PR #msc:60F17 #msc:60J55 #msc:60J60 #msc:62M05
paper · pdf · doi:10.1214/009117905000000152
published as Annals of Probability 2005, Vol. 33, No. 4, 1422-1451 · 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)
openalex publication_date 2005/07/01 · arxiv created 2005/07/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
We consider the empirical process \mathbbGt 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 \mathbbGt converge weakly to those of a zero-mean Gaussian random process \mathbbG. We prove that the weak convergence \mathbbGt⇒ \mathbbG takes place in ℓ∞(ℱ) if and only if the limit \mathbbG 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.