Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On U- and V-statistics for discontinuous Itô semimartingales

Mark Podolskij, Christian Schmidt, and Mathias Vetter

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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.


Dans cet article, nous étudions la théorie asymptotique de U-statistiques et de V-statistiques pour des semimartingales d’Itô discontinues qui sont observées à haute fréquence. Pour différents types de fonctions de noyaux, nous montrons des lois des grands nombres et des théorèmes de la limite centrale vers des lois stables. Dans la majorité des cas, le processus limite est conditionnellement centré Gaussien. La structure du noyau détermine si le la partie de sauts et/ou la partie continue de la semimartingale contribue à la limite.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1007-1050.

Received: 25 March 2015
Revised: 29 January 2016
Accepted: 7 February 2016
First available in Project Euclid: 21 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62F12: Asymptotic properties of estimators
Secondary: 60G48: Generalizations of martingales 60H05: Stochastic integrals

High frequency data Limit theorems Semimartingales Stable convergence U-statistics


Podolskij, Mark; Schmidt, Christian; Vetter, Mathias. On U- and V-statistics for discontinuous Itô semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1007--1050. doi:10.1214/16-AIHP744.

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