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The Wasserstein distance has been an attractive tool in many fields. But dueto its high computational complexity and the phenomenon of the curse ofdimensionality in empirical estimation, various extensions of the Wassersteindistance have been proposed to overcome the shortcomings such as the SlicedWasserstein distance. It enjoys a low computational cost and dimension-freesample complexity, but there are few distributional limit results of it. Inthis paper, we focus on Sliced 1-Wasserstein distance and its variantmax-Sliced 1-Wasserstein distance. We utilize the central limit theorem inBanach space to derive the limit distribution for the Sliced 1-Wassersteindistance. Through viewing the empirical max-Sliced 1-Wasserstein distance as asupremum of an empirical process indexed by some function class, we prove thatthe function class is P-Donsker under mild moment assumption. Moreover, forcomputing Sliced p-Wasserstein distance based on Monte Carlo method, we explorethat how many random projections that can make sure the error small in highprobability. We also provide upper bound of the expected max-Sliced1-Wasserstein between the true and the empirical probability measures underdifferent conditions and the concentration inequalities for max-Sliced1-Wasserstein distance are also presented. As applications of the theory, weutilize them for two-sample testing problem.

Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance