The Annals of Applied Probability

Limit theorems for nondegenerate U-statistics of continuous semimartingales

Mark Podolskij, Christian Schmidt, and Johanna F. Ziegel

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 6 (2014), 2491-2526.

Dates
First available in Project Euclid: 26 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1409058038

Digital Object Identifier
doi:10.1214/13-AAP983

Mathematical Reviews number (MathSciNet)
MR3262509

Zentralblatt MATH identifier
1308.60051

Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles
Secondary: 60G48: Generalizations of martingales 60H05: Stochastic integrals

Keywords
High frequency data limit theorems semimartingales stable convergence $U$-statistics

Citation

Podolskij, Mark; Schmidt, Christian; Ziegel, Johanna F. Limit theorems for nondegenerate U -statistics of continuous semimartingales. Ann. Appl. Probab. 24 (2014), no. 6, 2491--2526. doi:10.1214/13-AAP983. https://projecteuclid.org/euclid.aoap/1409058038


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