In this note, we investigate the convergence of a -statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and convergence and present some counter-examples showing that the -statistic itself might fail to converge: centering is needed as well as finiteness of .
Dans cette note, nous étudions le théorème ergodique pour des -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 ainsi que des contre-exemples montrant que la -statistique seule peut ne pas converger : un terme de centrage est requis ainsi que la finitude de .
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Herold Dehling 1; Davide Giraudo 2; Dalibor Volný 3
@article{CRMATH_2023__361_G9_1511_0, author = {Herold Dehling and Davide Giraudo and Dalibor Voln\'y}, title = {Some remarks on the ergodic theorem for $U$-statistics}, journal = {Comptes Rendus. Math\'ematique}, pages = {1511--1519}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.494}, language = {en}, }
TY - JOUR AU - Herold Dehling AU - Davide Giraudo AU - Dalibor Volný TI - Some remarks on the ergodic theorem for $U$-statistics JO - Comptes Rendus. Mathématique PY - 2023 SP - 1511 EP - 1519 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.494 LA - en ID - CRMATH_2023__361_G9_1511_0 ER -
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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