  In this paper we study multivariate ranks and quantiles, defined using the
theory of optimal transport, and build on the work of Chernozhukov et al.(2017)
and Hallin et al.(2021). 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. Under mild structural assumptions, we
provide global and local rates of convergence of the empirical quantile and
rank maps. We also provide a sub-Gaussian tail bound for the global L_2-loss of
the empirical quantile function. Further, we propose tuning parameter-free
multivariate nonparametric tests -- a two-sample test and a test for mutual
independence -- based on our notion of multivariate quantiles/ranks. Asymptotic
consistency of these tests are shown and the rates of convergence of the
associated test statistics are derived, both under the null and alternative
hypotheses.
