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In this paper, we study the asymptotic behavior of sums of functions of theincrements of a given semimartingale, taken along a regular grid whose meshgoes 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 theconvergence in probability of two types of such sums, and we also giveassociated central limit theorems. This extends known results when the summandsare a function depending only on the increments, and this is motivated mainlyby statistical applications.

Convergence of some random functionals of discretized semimartingales