Let $X_1,\dots,X_n$ be a set of independent samples of a distribution $\mu$ on the unit square, let $\hat\mu_n$ be the empirical distribution on the points $X_1,\dots,X_n$, and let $W_1(\mu,\hat\mu_n)$ denote the $1$-Wasserstein distance between $\hat\mu_n$ and $\mu$. Are there any simple, concrete bounds on $\Pr(W_1(\mu,\hat\mu_n)\geq t)$, where $n\leq 100$ and $t$ is fairly large (e.g. order of $n^{-1/2}$)?.

There are many results that describe the asymptotic behavior of this quantity, such as the papers http://www.emis.ams.org/journals/EJP-ECP/article/download/958/1147.pdf and http://link.springer.com/article/10.1007/s00440-014-0583-7 , but these tend to take the form of existence proofs and limiting statements (e.g. "There exist constants $C$ and $c$ such that $\Pr(W_1(\mu,\hat\mu_n)\geq t)\to (*)$). Are there any simple "ad-hoc" bounds that can bound $\Pr(W_1(\mu,\hat\mu_n)\geq t)$ for fixed values of $n$ and $t$?