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 $\mathrm{\Phi }$. 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 $\mathrm{\Phi }$ deviates by more than $\epsilon +\gamma (t)$ with probability at most $2e^{-2N{\epsilon }^2}+e^{-N{g}_{+}(t{)}^2}+e^{-N{g}_{-}(t{)}^2}$ 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.