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arXiv · 2512.24367

Limit theorems for the distance of random points in $l_p^n$-balls

David Alonso-Gutiérrez·Javier Martín Goñi·Joscha Prochno
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Abstract

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 $p \geq 2$.

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David Alonso-Gutiérrez, Javier Martín Goñi, Joscha Prochno. 2025-12-30. Limit theorems for the distance of random points in $l_p^n$-balls. https://arxiv.org/abs/2512.24367

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