This paper derives the asymptotic behavior of realized power variation of pure-jump Itô semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled Itô semimartingale over a fixed interval.
"Limit theorems for power variations of pure-jump processes with application to activity estimation." Ann. Appl. Probab. 21 (2) 546 - 588, April 2011. https://doi.org/10.1214/10-AAP700