Central limit theorems for martingales-I: Continuous limits

Abstract

When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient conditions for both processes to be independent. As examples of applications, we revisit some known results for the occupation times of Brownian motion and symmetric random walks. In the latter case, our proof is much simpler than the construction of strong approximations. Furthermore, we extend finite dimensional convergence of statistical estimators of financial volatility measures to convergence as stochastic processes.