  The estimation of local characteristics of Ito semimartingales has received a
great deal of attention in both academia and industry over the past decades. In
various papers limit theorems were derived for functionals of increments and
ranges in the infill asymptotics setting. In this paper we establish the
asymptotic theory for a wide class of statistics that are built from the
incremental process of an Ito semimartingale. More specifically, we will show
the law of large numbers and the associated stable central limit theorem for
the path dependent functionals in the continuous and discontinuous framework.
Some examples from economics and physics demonstrate the potential
applicability of our theoretical results in practice.
