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 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.

Published in at http://dx.doi.org/10.1214/10-AAP700 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Testing for jumps in a discretely observed process
• Estimating the degree of activity of jumps in high frequency data
• Testing for common arrivals of jumps for discretely observed multidimensional processes
• Maximum likelihood estimation for $\alpha$-stable autoregressive processes

Cited by in corpus (6)

• Efficient estimation of integrated volatility in presence of infinite variation jumps
• Testing for pure-jump processes for high-frequency data
• Jump activity estimation for pure-jump semimartingales via self-normalized statistics
• Realized Laplace transforms for pure-jump semimartingales
• Nonparametric inference on Lévy measures and copulas
• Limit theorems for the empirical distribution function of scaled increments of Itô semimartingales at high frequencies