An Exponential Concentration Inequality for the Components of a Uniform Random Vector on the Sphere

TL;DR

This paper establishes an exponential concentration inequality for the empirical distribution of scaled components of a uniform vector on the sphere, showing rapid convergence to Gaussian behavior with explicit bounds.

Contribution

It provides explicit, finite-sample concentration bounds for the empirical CDF of scaled sphere components, improving understanding of their Gaussian approximation.

Findings

Exponential decay of deviation probability in N

Explicit functions for bounds are derived

Finite-sample guarantees are provided

Abstract

We show that if $\vec X = (X_1, \dots, X_N)$ is a uniform random vector on the unit Euclidean sphere, the empirical CDF of the components of $\sqrt N \vec X = (\sqrt N X_1, \dots, \sqrt N X_N)$ concentrates exponentially rapidly in $N$ around the standard Gaussian CDF $\Phi$. More precisely, we find explicit functions $\gamma$ and $g_\pm$ such that the Kolmogorov-Smirnov distance between the empirical CDF of the components of $\sqrt N \vec X$ and $\Phi$ deviates by more than $\epsilon + \gamma(t)$ with probability at most $2e^{-2N\epsilon^2} + e^{-Ng_+(t)^2} + e^{-Ng_-(t)^2}$ for $\epsilon > 0$ and $t\in[0,1)$. A weaker but more transparent inequality replacing $\gamma$ and $g_\pm$ with linear functions is obtained as a corollary. All functions and constants are explicit, so our bounds offer finite-sample guarantees for statistical applications.