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

TL;DR

This work studies the fluctuations and rare-event behavior of the Euclidean distance between two independent random points in high-dimensional $l_p^n$-balls or their boundaries for $1\le p\le \infty$. It derives a precise central limit theorem (CLT) for the normalized distance, with explicit mean and variance in terms of $p$ via the Schechtman–Zinn representation and the delta method, and includes a separate CLT for the $\infty$-cube case. In addition, the paper establishes large deviation principles (LDPs) for the distance in the regimes $p\ge 2$, distinguishing between boundary and ball distributions as well as the cube, with rate functions expressed through Legendre–Fenchel transforms of a two-variable cumulant generating function. A compact sphere-case proof (Hammersley) is extended to general $l_p^n$-balls, highlighting the connection between high-dimensional geometry and probabilistic tail behavior and providing a complete picture of both typical fluctuations and rare events for these geometric objects.

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 $p \geq 2$.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: Schechtman and Zinn
• Lemma 2.2: Delta method
• Lemma 2.3: Cramer's theorem
• Lemma 2.4: Exponential equivalence