  In this paper, we study the asymptotic behavior of sums of functions of the
increments of a given semimartingale, taken along a regular grid whose mesh
goes to 0. The function of the $i$th increment may depend on the current time,
and also on the past of the semimartingale before this time. We study the
convergence in probability of two types of such sums, and we also give
associated central limit theorems. This extends known results when the summands
are a function depending only on the increments, and this is motivated mainly
by statistical applications.
