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For a separable metric space (X, d) Lp Wasserstein metrics between probability measures μ and v on X are defined by
\documentclass{article}\pagestyle{empty}\begin{document}$$ W_p \left({\mu,\nu } \right): = \inf \left\{ {\left({\mathop f\limits_{x \times x} d^p \left({x,y} \right)d\eta } \right)^{1/p} } \right\},p\;\varepsilon \;\left({1,\infty } \right),$$\end{document}
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.