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On $$U$$- and $$V$$-statistics for discontinuous Itô semimartingales. (English. French summary) Zbl 1372.60063
Summary: In this paper we examine the asymptotic theory for $$U$$-statistics and $$V$$-statistics of discontinuous Itô 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.
##### MSC:
 60G48 Generalizations of martingales 60F05 Central limit and other weak theorems 60H05 Stochastic integrals 62F12 Asymptotic properties of parametric estimators
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