Let $p\in[1,\infty)$ and let $d\in\mathbb N$ satisfy $d>2p$. Let $Q:=[0,1]^d$, equipped with the Borel $\sigma$-algebra $\mathcal B(Q)$ and the Euclidean distance inherited from $\mathbb R^d$. For Borel probability measures $\nu_0$ and $\nu_1$ on $(Q,\mathcal B(Q))$, let $\Pi(\nu_0,\nu_1)$ denote the set of Borel probability measures on $(Q\times Q,\mathcal B(Q)\otimes\mathcal B(Q))$ whose first marginal is $\nu_0$ and whose second marginal is $\nu_1$, and define

\begin{align*} W_p(\nu_0,\nu_1):=\inf_{\pi\in\Pi(\nu_0,\nu_1)}\left(\int_{Q\times Q}|x-y|^p\,d\pi(x,y)\right)^{1/p}. \end{align*}

Let $\mu$ be a Borel probability measure on $(Q,\mathcal B(Q))$ such that $\mu\ll \mathcal L^d\!\restriction_Q$. Write $\rho:Q\to[0,\infty)$ for the Radon--Nikodym derivative of $\mu$ with respect to $\mathcal L^d\!\restriction_Q$, so that $d\mu=\rho\,d(\mathcal L^d\!\restriction_Q)$. Assume that there exist constants $c,C\in(0,\infty)$ such that

\begin{align*} c\le \rho(x)\le C \end{align*}

for $(\mathcal L^d\!\restriction_Q)$-a.e. $x\in Q$. Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X_1,X_2,\dots$ be independent identically distributed random variables

\begin{align*} X_i:(\Omega,\mathcal F)\to(Q,\mathcal B(Q)) \end{align*}

with common law $\mu$. For every $N\in\mathbb N$, define the empirical measure $\hat\mu_N:\Omega\to\mathcal P(Q)$ by

\begin{align*} \hat\mu_N(\omega):=\frac{1}{N}\sum_{i=1}^N\delta_{X_i(\omega)}. \end{align*}

Then there exist constants $A,B\in(0,\infty)$, depending only on $d$, $p$, $c$, and $C$, such that, for every $N\in\mathbb N$,

\begin{align*} A N^{-1/d}\le \mathbb E[W_p(\hat\mu_N,\mu)]\le B N^{-1/d}. \end{align*}