ON A FORMULA FOR THE L2 WASSERSTEIN METRIC BETWEEN MEASURES ON EUCLIDEAN AND HILBERT-SPACES

For a separable metric space (X, d) Lp Wasserstein metrics between probability measures μ and v on X are defined by (Formula Presented.) where the infimum is taken over all probability measures η on X × X with marginal distributions μ and v, respectively. After mentioning some basic properties of these metrics as well as explicit formulae for X = R a formula for the L2 Wasserstein metric with X = Rn will be cited from [5], [9], and [21] and proved for any two probability measures of a family of elliptically contoured distributions. Finally this result will be generalized for Gaussian measures to the case of a separable Hilbert space. Copyright © 1990 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim