Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions

TL;DR

This paper proves a Central Limit Theorem for the volume of random partial coverings on high-dimensional spheres, showing Gaussian fluctuations and providing convergence rate bounds.

Contribution

It introduces a geometric analogue of the balls-into-bins problem and establishes a CLT for sphere coverings in both fixed and high-dimensional regimes.

Findings

Fluctuations of the covering volume are asymptotically Gaussian.

Quantitative bounds on convergence rate in Kolmogorov distance.

Results apply in fixed and high-dimensional regimes with dimension growing logarithmically.

Abstract

We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous geometric analogue of the classical balls-into-bins problem. We establish a Central Limit Theorem for the volume of the resulting random partial covering, showing that its fluctuations are asymptotically Gaussian. Moreover, we obtain a quantitative bound on the rate of convergence in the Kolmogorov distance. Our results hold both in fixed dimension and in a high-dimensional regime where the dimension grows at most logarithmically with $N$.