An exponential concentration inequality for the components of a uniform random vector on the sphere

We show that if X→=(X1,…,XN) is a uniform random vector on the unit Euclidean sphere, the empirical CDF of the components of NX→=( NX1,…, NXN) concentrates exponentially rapidly in N around the standard Gaussian CDF Φ. More precisely, we find explicit functions γ and g± such that the Kolmogorov-Smirnov distance dKS between the empirical CDF of the components of NX→ and Φ deviates by more than ϵ+γ(t) with probability at most 2e−2Nϵ2+e−Ng+(t)2+e−Ng−(t)2 for ϵ>0 and t∈[0,1). A weaker but more transparent corollary yields the explicit sub-gaussian deviation bound PrdKS(FˆNX→,Φ)>δ≤4e−964Nδ2,δ>0 All functions and constants are explicit, so our bounds offer finite-sample guarantees for statistical applications.