Some remarks on the ergodic theorem for $U$-statistics
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 neede...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.494/ |
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| author | Dehling, Herold Giraudo, Davide Volný, Dalibor |
| author_facet | Dehling, Herold Giraudo, Davide Volný, Dalibor |
| author_sort | Dehling, Herold |
| collection | DOAJ |
| description | 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 $\sup _{j\ge 2}\mathbb{E}[|h(X_1,X_j)|]$. |
| format | Article |
| id | doaj-art-3956bb636b744c2184af4006c2534e2b |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3956bb636b744c2184af4006c2534e2b2025-02-07T11:10:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G91511151910.5802/crmath.49410.5802/crmath.494Some remarks on the ergodic theorem for $U$-statisticsDehling, Herold0Giraudo, Davide1Volný, Dalibor2Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, GermanyInstitut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg and CNRS 7 rue René Descartes 67000 Strasbourg, FranceUniversity de Rouen, LMRS and CNRS UMR 6085.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 $\sup _{j\ge 2}\mathbb{E}[|h(X_1,X_j)|]$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.494/ |
| spellingShingle | Dehling, Herold Giraudo, Davide Volný, Dalibor Some remarks on the ergodic theorem for $U$-statistics Comptes Rendus. Mathématique |
| title | Some remarks on the ergodic theorem for $U$-statistics |
| title_full | Some remarks on the ergodic theorem for $U$-statistics |
| title_fullStr | Some remarks on the ergodic theorem for $U$-statistics |
| title_full_unstemmed | Some remarks on the ergodic theorem for $U$-statistics |
| title_short | Some remarks on the ergodic theorem for $U$-statistics |
| title_sort | some remarks on the ergodic theorem for u statistics |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.494/ |
| work_keys_str_mv | AT dehlingherold someremarksontheergodictheoremforustatistics AT giraudodavide someremarksontheergodictheoremforustatistics AT volnydalibor someremarksontheergodictheoremforustatistics |