Limit theorems for moving averages of discretized processes plus noise

TL;DR

This paper develops limit theorems for functionals of high-frequency observed moving averages of semimartingales with noise, extending pre-averaging methods to provide consistent estimates and multidimensional CLTs.

Contribution

It generalizes the pre-averaging approach to handle more complex semimartingale functionals with noise, establishing new limit theorems and convergence rates.

Findings

Established limit theorems for functionals of discretized semimartingales with noise.

Proved multidimensional stable central limit theorems with rate n^{-1/4}.

Provided consistent estimators for characteristics of general semimartingales.

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.