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

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.