  In this work we introduce a novel approach of construction of multivariate
cumulative distribution functions, based on cyclical-monotone mapping of an
original measure $\mu \in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ to some target
measure $\nu \in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ , supported on a convex
compact subset of $\mathbb{R}^d$. This map is referred to as $\nu$-Brenier
distribution function ($\nu$-BDF), whose counterpart under the one-dimensional
setting $d = 1$ is an ordinary CDF, with $\nu$ selected as $\mathcal{U}[0, 1]$,
a uniform distribution on $[0, 1]$. Following one-dimensional frame-work, a
multivariate analogue of Glivenko-Cantelli theorem is provided. A practical
applicability of the theory is then illustrated by the development of a
non-parametric pivotal two-sample test, that is rested on $2$-Wasserstein
distance.
