In this paper we study multivariate ranks and quantiles, defined using the theory of optimal transportation, and build on the work of Chernozhukov et al. (2017) and del Barrio et al. (2018). We study the characterization, computation and properties of the multivariate rank and quantile functions and their empirical counterparts. We derive the uniform consistency of these empirical estimates to their population versions, under certain assumptions. In fact, we prove a Glivenko-Cantelli type theorem that shows the asymptotic stability of the empirical rank map in any direction. We provide easily verifiable sufficient conditions that guarantee the existence of a continuous and invertible population quantile map --- a crucial assumption for our main consistency result. We provide a framework to derive the local uniform rate of convergence of the estimated quantile and ranks functions and explicitly illustrate the technique in a special case. Further, we propose multivariate (nonparametric) goodness-of-fit tests --- a two-sample test and a test for mutual independence --- based on our notion of quantiles and ranks. Asymptotic consistency of these tests are also shown. Additionally, we derive many properties of (sub)-gradients of convex functions and their Legendre-Fenchel duals that may be of independent interest.
View on arXiv