When a distribution is observed through samples, its empirical measure is the canonical plug-in estimator in Wasserstein space. Consistency is simple, but rates depend sharply on dimension, moment assumptions, and the local structure of the underlying law.
Empirical measures
Definition. For i.i.d. \(X_1,\ldots,X_n\sim\mu\), \[ \widehat\mu_n=\frac1n\sum_{i=1}^n\delta_{X_i}. \] Its integral against \(f\) is the empirical average \(\widehat\mu_nf=n^{-1}\sum_i f(X_i)\).
Strong Wasserstein consistency. If \(\mu\in\mathcal P_p(\mathbb R^d)\), then \[ W_p(\widehat\mu_n,\mu)\longrightarrow0 \qquad\text{almost surely}. \]
Proof. Varadarajan's theorem, or the strong law on a countable convergence-determining class, gives \(\widehat\mu_n\Rightarrow\mu\) almost surely. The scalar strong law gives \[ \int|x|^p\,\mathrm{d}\widehat\mu_n(x) =\frac1n\sum_{i=1}^n|X_i|^p \longrightarrow\mathbb E|X_1|^p. \] The topology theorem on page 11 converts weak convergence plus moment convergence into \(W_p\)-convergence.
Dimension-dependent expected rates
Generic compact-support rates. Suppose \(\mu\) is supported in a bounded subset of \(\mathbb R^d\). Up to constants depending on \(d,p\) and the support diameter, \[ \mathbb E W_p^p(\widehat\mu_n,\mu) \lesssim \begin{cases} n^{-1/2},&d<2p,\\ n^{-1/2}\log(1+n),&d=2p,\\ n^{-p/d},&d>2p. \end{cases} \] Comparable bounds hold under sufficiently high finite moments, with additional tail terms.
Proof mechanism. Partition space into dyadic cubes. A transport plan first corrects mass discrepancies on coarse cubes, then recursively within each cube. The cost at scale \(2^{-j}\) is controlled by \(2^{-jp}\) times the total empirical mass fluctuation over the \(2^{jd}\) cells. Binomial fluctuations contribute roughly \(\min\{1,\sqrt{2^{jd}/n}\}\). Summing \[ \sum_{j\ge0}2^{-jp}\min\left\{1,\sqrt{\frac{2^{jd}}n}\right\} \] yields the three regimes. For unbounded support, apply the argument on dyadic annuli and control the remaining tail with moments.
These are distribution-free rates and need not be sharp for smooth low-dimensional laws. They expose the central obstruction: at fine scales there are exponentially many cells relative to the dimension.
Why the curse of dimensionality appears
Quantization lower bound. If \(\mu\) has a density bounded below on a set of positive \(d\)-dimensional volume, then every probability measure \(\nu_n\) supported on at most \(n\) points satisfies \[ W_p(\mu,\nu_n)\ge c\,n^{-1/d}. \]
Proof. Let the support points be \(z_1,\ldots,z_n\). For any transport to these points, each \(x\) must move at least \(\min_i|x-z_i|\), so \[ W_p^p(\mu,\nu_n)\ge\int\min_i|x-z_i|^p\,\mathrm{d}\mu(x). \] The union of \(n\) balls of radius \(r\) has volume at most \(n\omega_dr^d\). Choose \(r\asymp n^{-1/d}\) so these balls cover at most half of a fixed positive-volume region. On the uncovered part the distance is at least \(r\), giving the lower bound.
Since \(\widehat\mu_n\) has at most \(n\) support points, no estimator constrained to the raw empirical support can beat this geometric approximation barrier uniformly over full-dimensional distributions.
Discrete and low-dimensional exceptions
Finite support. If \(\mu\) is supported on a fixed finite set with pairwise distances bounded above and below, then \(W_p^p(\widehat\mu_n,\mu)\) is comparable to the \(\ell^1\) error of the empirical cell probabilities, which is \(O_{\mathbb P}(n^{-1/2})\). The effective dimension is the number of atoms, not the ambient Euclidean dimension.
One dimension. The identity \[ W_p^p(\widehat\mu_n,\mu) =\int_0^1|F_n^{-1}(u)-F^{-1}(u)|^p\,\mathrm{d}u \] reduces the problem to quantile-process theory. If \(F\) has a positive smooth density on a compact interval, \(W_p(\widehat\mu_n,\mu)=O_{\mathbb P}(n^{-1/2})\), much faster than the worst-case bound for \(W_p\) obtained by taking a \(p\)-th root above.
Concentration around the mean
Bounded-support concentration. If \(\mu\) has support diameter \(D\), changing one observation changes \(W_p^p(\widehat\mu_n,\mu)\) by at most \(D^p/n\). Therefore McDiarmid's inequality gives \[ \mathbb P\left( W_p^p(\widehat\mu_n,\mu) -\mathbb EW_p^p(\widehat\mu_n,\mu)\ge t \right) \le\exp\left(-\frac{2nt^2}{D^{2p}}\right). \]
Proof. If \(X_i\) is replaced by \(X_i'\), couple the two empirical measures by matching all unchanged observations and moving only mass \(1/n\) from \(X_i\) to \(X_i'\). Their \(W_p^p\) distance is at most \(D^p/n\). The triangle inequality for optimal values, or direct replacement in a coupling to \(\mu\), gives the same bounded difference for the statistic. Apply McDiarmid's inequality.
Two-sample empirical transport
For independent samples with laws \(\mu,\nu\), \[ |W_p(\widehat\mu_n,\widehat\nu_m)-W_p(\mu,\nu)| \le W_p(\widehat\mu_n,\mu)+W_p(\widehat\nu_m,\nu). \] Thus one-sample rates immediately imply consistency and deviation bounds away from the null. Under \(\mu=\nu\), however, the first-order derivative of the distance may vanish and the limiting scale can change. Page 21 treats this nonregular behavior.
Tail assumptions are structural
A small empirical error in weak topology can hide a large transport error in the tails. Finite \(p\)-th moment is sufficient for consistency but generally not for a clean finite-sample rate. Polynomial moments yield polynomial tail terms, exponential moments yield exponential concentration, and compact support removes tail error entirely. Any statistical claim about empirical Wasserstein distance must therefore state both dimension and tail assumptions.