Limit theorems for nondegenerate U-statistics of continuous semimartingales

TL;DR

This paper develops asymptotic theory for nondegenerate U-statistics based on high-frequency data from continuous semimartingales, establishing convergence and a stable CLT with a Gaussian limit, with potential statistical applications.

Contribution

It introduces a new asymptotic framework for U-statistics of continuous semimartingales, including uniform convergence and a stable CLT with a Gaussian limit.

Findings

Proves uniform convergence in probability.

Establishes a functional stable central limit theorem.

Shows the limiting process is conditionally Gaussian.

Abstract

This paper presents the asymptotic theory for nondegenerate $U$-statistics of high frequency observations of continuous It\^{o} 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.