Asymptotics for L2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances

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Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications (2017, background): “In a similar spirit del Barrio et al. (2005) provided asymptotics for a weighted version of the empirical 2-Wasserstein distance in one dimension and Freitag and Munk (2005) derive limit laws for semiparametric models, still restricted to the univariate case.”
Central limit theorems for empirical transportation cost in general dimension (2017, background): “If p > 1 then it is still possible to get a limiting distribution for √ nWp(Pn, P ), but now integrability assumptions are not enough and the available results require some smoothness conditions on P (and on its density), see [5] for the case p = 2.”
Inference for empirical Wasserstein distances on finite spaces (2016, background): “In the context of deformable models distributional results have been proved (Del Barrio et al., 2015) for specific multi-dimensional parametric models which factor into one-dimensional parts.”
Statistical Aspects of Wasserstein Distances (2018, background): “In the context of Wasserstein distance, del Barrio et al. (2005) study the limiting behaviour of the norm (cid:107) q n 22 ,w = 1 0 q 2 n ( t ) w ( t ) d t , for an integrable weight function w on (0 , 1).”