Does 1-Wasserstein metric have bounds?

jer

New Member
Given $$l_1(u,v)=\int_{-\infty}^{+\infty}|U-V|$$ the 1-Wasserstein distance with $$u$$ and $$v$$ two probability distributions and $$U$$ and $$V$$ their respective CDFs. The values of $$u$$ and $$v$$ are all positive in my case.

Let's assume that I have already computed $$a=max(u)$$, $$b=max(v)$$, $$c=min(u)$$, and $$d=min(v)$$,
Does $$l_1$$have an upper and a lower bound?

jer

New Member
Yes it's true, what's about the upper bound?

fed2

Active Member
Yes it's true, what's about the upper bound?
Im going 'no' since the U,V values at mins or maxes could be as far apart as you like.

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