Comptes Rendus
Probability theory, Statistics
Some remarks on the ergodic theorem for U-statistics
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1511-1519.

In this note, we investigate the convergence of a U-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and L 1 convergence and present some counter-examples showing that the U-statistic itself might fail to converge: centering is needed as well as finiteness of sup j2 𝔼[|h(X 1 ,X j )|].

Dans cette note, nous étudions le théorème ergodique pour des U-statisques d’ordre 2 dont les données sont issues d’une suite strictement stationnaire. Nous présentons des conditions suffisantes pour la convergence presque sûre et dans L 1 ainsi que des contre-exemples montrant que la U-statistique seule peut ne pas converger : un terme de centrage est requis ainsi que la finitude de sup j2 𝔼[|h(X 1 ,X j )|].

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.494
Classification: 37A30, 60F05

Herold Dehling 1; Davide Giraudo 2; Dalibor Volný 3

1 Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
2 Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg and CNRS 7 rue René Descartes 67000 Strasbourg, France
3 University de Rouen, LMRS and CNRS UMR 6085.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Herold Dehling; Davide Giraudo; Dalibor Volný. Some remarks on the ergodic theorem for $U$-statistics. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1511-1519. doi : 10.5802/crmath.494. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.494/

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