Convergence of Multivariate Quantile Surfaces

We define the quantile set of order α ∈ [1/2, 1) associated to a law P on R d to be the collection of its directional quantiles seen from an observer O ∈ R d. Under minimal assumptions these star-shaped sets are closed surfaces, continuous in (O, α) and the collection of empirical quantile surfaces is uniformly consistent. Under mild assumptions – no density or symmetry is required for P – our uniform central limit theorem reveals the correlations between quantile points and a non asymptotic Gaussian approximation provides joint confident enlarged quantile surfaces. Our main result is a dimension free rate n −1/4 (log n) 1/2 (log log n) 1/4 of Bahadur-Kiefer embedding by the empirical process indexed by half-spaces. These limit theorems sharply generalize the univariate quantile convergences and fully characterize the joint behavior of Tukey half-spaces