Rates of convergence to the local time of Oscillating and Skew Brownian Motions

In this paper a class of statistics based on high frequency observations of oscillating Brownian motions and skew Brownian motions is considered. Their convergence rate towards the local time of the underling process is obtained in form of a Central Limit Theorem. The convergence speed is the same as in the case of Brownian motion but oscillating and skew Brownian motion have singular coecients and their local time is discontinuous.