Limit theorems for power variations of pure-jump processes with application to activity estimation

Abstract

This paper derives the asymptotic behavior of realized power variation of pure-jump It\^{o} 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 It\^{o} semimartingale over a fixed interval.