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How is the lower Frechet-Hoeffding copula bound proved?

In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$.

I'm struggling to prove it in higher dimensions.

Peter_Pan
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user62764
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2 Answers2

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Note that copulae are uniformly continuous with variation bound $$ |C(x)-C(y)|\leq \|x-y\|_1=\sum_{i=1}^n |x_i-y_i|. $$

Using this result we get $$1-C(x)=|C(1,1,...,1)-C(x)| \leq \sum_{i=1}^n |1-x_i|=n-\sum_{i=1}^n x_i$$ which can be rearranged to $$ \sum_{i=1}^n x_i -n+1 \leq C(x)$$ as required.

The proof of the variation bound itself can be found for instance in "Probabilistic metric spaces" (1983) by Schweizer and Sklar.

ondrejb
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The lower bound is only valid in 2 dimension. That's why you can't prove it in higher dimension.

user133034
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