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Given $u$ and $v$ two probability distributions and U and V their respective $CDFs$, the $1$-Wasserstein distance is formulated as follows:

$l_1(u,v)=\int_{-\infty}^{+\infty}|U-V|$

Does $l_1$ have an upper and a lower bound? Note that the values of u et v are all positive in my case and I have $max(u)$, $max(v)$, $min(u)$, and $min(v)$ already calculated.

Thank you in advance for your clarification.

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A lower bound is zero. It is attained when $U=V$ (up to a set of zero measure).

An upper bound is two, since $$ \int|U-V| \leq \int|U|+\int|V| = \int U+\int V = 2, $$ since both $U$ and $V$ are probability distributions. The upper bound is attained, e.g., when $U$ and $V$ have all their probability on disjoint sets.

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  • $\begingroup$ Thank you for your response. In the case where U and V are CDFs of u and v with u and v as probability distributions or measures extracted from experiments for example, what is the upper bound of $l_1(u,v)$? $\endgroup$
    – curiosus
    Jun 7 '20 at 8:53

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