Let $ \mu $ and $ \nu $ be two probability distributions, and $ \hat{\mu}_n $ and $ \hat{\nu}_n$ be their finite samples. Let $ W_2 $ denote the Wasserstein-2 distance and $ \hat{W_2} $ a numerical solver for $ W_2$ , say by linear programming method. I’m interested in estimating $ W_2( \mu, \nu ) $ from $ \hat{W_2}( \hat{\mu}_n, \hat{\nu}_n ) $ .

I’m wondering if there are any known results about the convergence of $ \hat{W_2} $ to $ W_2 $ from finite samples (note this is different from $ W_2( \hat{\mu}_n, \hat{\nu}_n ) \rightarrow W_2( \mu, \nu ) $ ). For example, is it true that for any $ \epsilon > 0 $ , $ \delta > 0 $ there exists $ N$ such that for any $ n > N $ , we have:

$ $ \mathbb{P}( | \hat{W_2}( \hat{\mu}, \hat{\nu} ) – W_2( \mu, \nu )| < \epsilon) \geq 1- \delta ?$ $ Or $ \hat{W_2}( \hat{\mu}, \hat{\nu} ) $ converges to $ W_2( \mu, \nu ) $ in distribution?