Cyclic and alternating $U$-statistics
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912646109855744 |
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| author | Janson, Svante |
| author_facet | Janson, Svante |
| contents | We define cyclic $U$-statistics as a variant of $U$-statistics based on variables $X_1,\dots,X_n$ that are assumed to be cyclically ordered. We also define alternating $U$-statistics where in the definition terms are summed with alternating sings (in three different ways). Only $U$-statistics of order 2 are considered. The definitions are inspired by special cases studied by Chebikin (2008) and Even-Zohar (2017) for random permutations.
We show limit theorems similar to well-known results for standard $U$-statistics, but with some differences between the different versions. In particular, we find both ``nondegenerate'' normal limits and ``degenerate'' non-normal limits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cyclic and alternating $U$-statistics Janson, Svante Probability 60F05, 60C05 We define cyclic $U$-statistics as a variant of $U$-statistics based on variables $X_1,\dots,X_n$ that are assumed to be cyclically ordered. We also define alternating $U$-statistics where in the definition terms are summed with alternating sings (in three different ways). Only $U$-statistics of order 2 are considered. The definitions are inspired by special cases studied by Chebikin (2008) and Even-Zohar (2017) for random permutations. We show limit theorems similar to well-known results for standard $U$-statistics, but with some differences between the different versions. In particular, we find both ``nondegenerate'' normal limits and ``degenerate'' non-normal limits. |
| title | Cyclic and alternating $U$-statistics |
| topic | Probability 60F05, 60C05 |
| url | https://arxiv.org/abs/2510.12480 |