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The Wasserstein distance is a metric on a space of probability measures thathas seen a surge of applications in statistics, machine learning, and appliedmathematics. However, statistical aspects of Wasserstein distances arebottlenecked by the curse of dimensionality, whereby the number of data pointsneeded to accurately estimate them grows exponentially with dimension. Gaussiansmoothing was recently introduced as a means to alleviate the curse ofdimensionality, giving rise to a parametric convergence rate in any dimension,while preserving the Wasserstein metric and topological structure. Tofacilitate valid statistical inference, in this work, we develop acomprehensive limit distribution theory for the empirical smooth Wassersteindistance. The limit distribution results leverage the functional delta methodafter embedding the domain of the Wasserstein distance into a certain dualSobolev space, characterizing its Hadamard directional derivative for the dualSobolev norm, and establishing weak convergence of the smooth empirical processin the dual space. To estimate the distributional limits, we also establishconsistency of the nonparametric bootstrap. Finally, we use the limitdistribution theory to study applications to generative modeling via minimumdistance estimation with the smooth Wasserstein distance, showing asymptoticnormality of optimal solutions for the quadratic cost.

Limit distribution theory for smooth $p$-Wasserstein distances