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In this paper, we consider a probabilistic setting where the probabilitymeasures are considered to be random objects. We propose a procedure ofconstruction non-asymptotic confidence sets for empirical barycenters in2-Wasserstein space and develop the idea further to construction of anon-parametric two-sample test that is then applied to the detection ofstructural breaks in data with complex geometry. Both procedures mainly rely onthe idea of multiplier bootstrap (Spokoiny and Zhilova (2015), Chernozhukov etal. (2014)). The main focus lies on probability measures that have commutingcovariance matrices and belong to the same scatter-location family: we proofthe validity of a bootstrap procedure that allows to compute confidence setsand critical values for a Wasserstein-based two-sample test.

Construction of Non-asymptotic Confidence Sets in 2-Wasserstein Space