Some remarks on the ergodic theorem for $U$-statistics
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913214130814976 |
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| author | Dehling, Herold G. Giraudo, Davide Volny, Dalibor |
| author_facet | Dehling, Herold G. Giraudo, Davide Volny, Dalibor |
| contents | 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 boundedness of $\sup_{j\geq 2}\mathbb{E}[|h(X_1,X_j)|]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04539 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some remarks on the ergodic theorem for $U$-statistics Dehling, Herold G. Giraudo, Davide Volny, Dalibor Probability 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 boundedness of $\sup_{j\geq 2}\mathbb{E}[|h(X_1,X_j)|]$. |
| title | Some remarks on the ergodic theorem for $U$-statistics |
| topic | Probability |
| url | https://arxiv.org/abs/2302.04539 |