  The smooth 1-Wasserstein distance (SWD) $W_1^\sigma$ was recently proposed as
a means to mitigate the curse of dimensionality in empirical approximation
while preserving the Wasserstein structure. Indeed, SWD exhibits parametric
convergence rates and inherits the metric and topological structure of the
classic Wasserstein distance. Motivated by the above, this work conducts a
thorough statistical study of the SWD, including a high-dimensional limit
distribution result for empirical $W_1^\sigma$, bootstrap consistency,
concentration inequalities, and Berry-Esseen type bounds. The derived
nondegenerate limit stands in sharp contrast with the classic empirical $W_1$,
for which a similar result is known only in the one-dimensional case. We also
explore asymptotics and characterize the limit distribution when the smoothing
parameter $\sigma$ is scaled with $n$, converging to $0$ at a sufficiently slow
rate. The dimensionality of the sampled distribution enters empirical SWD
convergence bounds only through the prefactor (i.e., the constant). We provide
a sharp characterization of this prefactor's dependence on the smoothing
parameter and the intrinsic dimension. This result is then used to derive new
empirical convergence rates for classic $W_1$ in terms of the intrinsic
dimension. As applications of the limit distribution theory, we study
two-sample testing and minimum distance estimation (MDE) under $W_1^\sigma$. We
establish asymptotic validity of SWD testing, while for MDE, we prove
measurability, almost sure convergence, and limit distributions for optimal
estimators and their corresponding $W_1^\sigma$ error. Our results suggest that
the SWD is well suited for high-dimensional statistical learning and inference.
