  This article is dedicated to the estimation of Wasserstein distances and
Wasserstein costs between two distinct continuous distributions $F$ and $G$ on
$\mathbb R$. The estimator is based on the order statistics of (possibly
dependent) samples of $F$ resp. $G$. We prove the consistency and the
asymptotic normality of our estimators. \begin{it}Keywords:\end{it} Central
Limit Theorems- Generelized Wasserstein distances- Empirical processes- Strong
approximation- Dependent samples.
