no. G9
Probabilités, Statistiques
Some remarks on the ergodic theorem for U-statistics
[Quelques remarques sur le théorème ergodique pour les U-statistiques]
Dehling, Herold1 ; Giraudo, Davide2  ; Volný, Dalibor3
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.
Comptes Rendus. Mathématique, Tome 361 (2023) no. G9, 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 j≥2 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 j≥2 𝔼[|h(X 1 ,X j )|].

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.494
Classification : 37A30, 60F05

Dehling, Herold 1 ; Giraudo, Davide 2 ; Volný, Dalibor 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.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Dehling, Herold; Giraudo, Davide; Volný, Dalibor. Some remarks on the ergodic theorem for $U$-statistics. Comptes Rendus. Mathématique, Tome 361 (2023) no. G9, pp. 1511-1519. doi: 10.5802/crmath.494

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