  The Wasserstein distance is an attractive tool for data analysis but
statistical inference is hindered by the lack of distributional limits. To
overcome this obstacle, for probability measures supported on finitely many
points, we derive the asymptotic distribution of empirical Wasserstein
distances as the optimal value of a linear program with random objective
function. This facilitates statistical inference (e.g. confidence intervals for
sample based Wasserstein distances) in large generality. Our proof is based on
directional Hadamard differentiability. Failure of the classical bootstrap and
alternatives are discussed. The utility of the distributional results is
illustrated on two data sets.
