Limit theorems for moving averages of discretized processes plus noise

Abstract

This paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency. Our method generalizes the pre-averaging approach (see [Bernoulli 15 (2009) 634--658, Stochastic Process. Appl. 119 (2009) 2249--2276]) and provides consistent estimates for various characteristics of general semimartingales. Furthermore, we prove the associated multidimensional (stable) central limit theorems. As expected, we find central limit theorems with a convergence rate $n^{-1/4}$, if $n$ is the number of observations.

Published in at http://dx.doi.org/10.1214/09-AOS756 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Testing for jumps in a discretely observed process
• Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps
• Estimating the degree of activity of jumps in high frequency data
• Are volatility estimators robust with respect to modeling assumptions?

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