Abstract

Wasserstein distances provide a metric on a space of probability measures. We consider the space $\Omega$ of all probability measures on the finite set $\chi =\left\{1,\dots ,n\right\}$, where $n$ is a positive integer. The 1-Wasserstein distance, $W_1\left(\mu ,\nu \right)$, is a function from $\Omega \times \Omega$ to $\left[0,\infty \right)$. This paper derives closed-form expressions for the first and second moments of $W_1$ on $\Omega \times \Omega$ assuming a uniform distribution on $\Omega \times \Omega$.

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Mathematical Subject Classification 2010

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