Limit theorems for the distance of random points in $l_p^n$-balls
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915700900102144 |
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| author | Alonso-Gutiérrez, David Goñi, Javier Martín Prochno, Joscha |
| author_facet | Alonso-Gutiérrez, David Goñi, Javier Martín Prochno, Joscha |
| contents | In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1 \leq p \leq \infty)$ or on its boundary satisfies a central limit theorem as $n$ tends to $\infty$. Also, we give a compact proof of the case of the sphere, which was proved by Hammersley. Furthermore, we complement our central limit theorem by providing large deviation principles for the cases $p \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit theorems for the distance of random points in $l_p^n$-balls Alonso-Gutiérrez, David Goñi, Javier Martín Prochno, Joscha Probability In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1 \leq p \leq \infty)$ or on its boundary satisfies a central limit theorem as $n$ tends to $\infty$. Also, we give a compact proof of the case of the sphere, which was proved by Hammersley. Furthermore, we complement our central limit theorem by providing large deviation principles for the cases $p \geq 2$. |
| title | Limit theorems for the distance of random points in $l_p^n$-balls |
| topic | Probability |
| url | https://arxiv.org/abs/2512.24367 |