# Does the 1-Wasserstein distance have an upper and a lower bound?

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.

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.
• 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 $l_1(u,v)$? Jun 7 '20 at 8:53