Podolskij, Mark et al., Limit theorems for nondegenerate U-statistics of continuous semimartingales, Annals of applied probability , 24(6), 2014, pp. 2491-2526
This paper presents the asymptotic theory for nondegenerate U-statistics of high frequency observations of continuous Itô semimartingales. We prove uniform convergence in probability and show a functional stable central limit theorem for the standardized version of the U-statistic. The limiting process in the central limit theorem turns out to be conditionally Gaussian with mean zero. Finally, we indicate potential statistical applications of our probabilistic results.