  The Wasserstein distance is a metric on a space of probability measures that
has seen a surge of applications in statistics, machine learning, and applied
mathematics. However, statistical aspects of Wasserstein distances are
bottlenecked by the curse of dimensionality, whereby the number of data points
needed to accurately estimate them grows exponentially with dimension. Gaussian
smoothing was recently introduced as a means to alleviate the curse of
dimensionality, giving rise to a parametric convergence rate in any dimension,
while preserving the Wasserstein metric and topological structure. To
facilitate valid statistical inference, in this work, we develop a
comprehensive limit distribution theory for the empirical smooth Wasserstein
distance. The limit distribution results leverage the functional delta method
after embedding the domain of the Wasserstein distance into a certain dual
Sobolev space, characterizing its Hadamard directional derivative for the dual
Sobolev norm, and establishing weak convergence of the smooth empirical process
in the dual space. To estimate the distributional limits, we also establish
consistency of the nonparametric bootstrap. Finally, we use the limit
distribution theory to study applications to generative modeling via minimum
distance estimation with the smooth Wasserstein distance, showing asymptotic
normality of optimal solutions for the quadratic cost.
