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

TL;DR

This paper establishes limit theorems for the realized power variation of pure-jump processes, enabling more accurate activity estimation of stochastic processes from high-frequency data.

Contribution

It introduces new asymptotic results for realized power variation and develops an adaptive estimator for process activity based on these theorems.

Findings

Proves convergence in probability of realized power variation.

Establishes a central limit theorem for realized power variation.

Proposes an efficient adaptive activity estimator.

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.