Before revision:  As an example of how to apply the limit theory provided in this paper, we build a time-varying friction parameter extension of the (semiparametric) model with uncertainty zones (Robert and Rosenbaum (2012)) and we show that we can verify the conditions for the estimation of integrated volatility. Moreover, practical applications in time series, such as the optimal block length and local bias-correction, are discussed and numerical simulations are carried on the local MLE of a time-varying parameter MA(1) model to illustrate them .
After revision:  As an example of how to apply the limit theory provided in this paper, we build a time-varying friction parameter extension of the specific convergence rate n^{1/2 time-varying parameter MA(1) model to illustrate them .