  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.
