  The typical central limit theorems in high-frequency asymptotics for
semimartingales are results on stable convergence to a mixed normal limit with
an unknown conditional variance. Estimating this conditional variance usually
is a hard task, in particular when the underlying process contains jumps. For
this reason, several authors have recently discussed methods to automatically
estimate the conditional variance, i.e. they build a consistent estimator from
the original statistics, but computed at various different time scales. Their
methods work in several situations, but are essentially restricted to the case
of continuous paths always. The aim of this work is to present a new method to
consistently estimate the conditional variance which works regardless of
whether the underlying process is continuous or has jumps. We will discuss the
case of power variations in detail and give insight to the heuristics behind
the approach.
