Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions

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 $N$.

11 pages, 1 figure