Second-order properties of thresholded realized power variations of FJA additive processes

Within a type of additive processes of finite jump activity (FJA), we give precise conditions for the mean-squared consistency and the validity of a feasible Central Limit Theorem for thresholded realized power variation estimators (TRV). The obtained results show that some conditions commonly used in the literature are rather strong and far from being necessary. To justify that the proposed conditions are the “best possible”, we also show that these are necessary for FJA Lévy processes. Our approach yields both non-asymptotic upper bounds and asymptotic decompositions of the mean-squared errors of the considered estimators. For comparison purposes, we also obtain the analog asymptotic decomposition for a general multi-power realized variation, which theoretically illustrates the numerically observed slow rate of convergence of its bias compared to that of an appropriately chosen TRV.