Estimation of volatility functionals: the case of a square root n window

We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency 1/\Delta_n, with \Delta_n going to zero. The goal of this paper is to provide an estimator for the integral over [0,t] of a given function of the volatility matrix, with the optimal rate 1/\sqrt{\Delta_n} and minimal asymptotic variance. To achieve this we use spot volatility estimators based on observations within time intervals of length k_n\Delta_n. In [5] this was done with k_n tending to infinity and k_n\sqrt{\Delta_n} tending to 0, and a central limit theorem was given after suitable de-biasing. Here we do the same with k_n of order 1/\sqrt{\Delta_n}. This results in a smaller bias, although more difficult to eliminate.