The paper derives a functional central limit theorem for the empirical distributions of a system of strongly correlated continuous martingales at the level of the full trajectory space. We provide a general class of functionals for which the weak convergence to a centered Gaussian random field takes place. An explicit formula for the covariance is established and a characterization of the limit is given in terms of an inductive system of SPDEs. We also show a density theorem for a Sobolev-type class of functionals on the space of continuous functions. © 2003 Elsevier SAS. All rights reserved.
Grigorescu, I. (2004). An infinite dimensional central limit theorem for correlated martingales. Annales de l’institut Henri Poincare (B) Probability and Statistics, 40(2), 167–196. https://doi.org/10.1016/j.anihpb.2003.03.001