  In this paper we examine the asymptotic theory for U-statistics and
V-statistics of discontinuous Ito semimartingales that are observed at high
frequency. For different types of kernel functions we show laws of large
numbers and associated stable central limit theorems. In most of the cases the
limiting process will be conditionally centered Gaussian. The structure of the
kernel function determines whether the jump and/or the continuous part of the
semimartingale contribute to the limit.
