# “Smallness” of the set of Archimedean copulas in the set of all exchangeable copulas

This is a little bit of a vague question I guess. Let us for simplicity consider bivariate copulas. We can think aboutset $\mathcal{C}$ of all exchangeable copulas: that is, these are copulas $C: [0,1]^2\rightarrow [0,1]$ that satisfy the following condition: $$C(u,v)=C(v,u).$$ We can also consider a subset $\mathcal{C}_0 \subset \mathcal{C}$ of Archimedean copulas $-$ these are copulas represented in the following form: $$C(u,v)=\phi^{[-1]}(\phi(u)+\phi(v)), \text{ where }$$ the generator $\phi:[0,1] \rightarrow [0,\infty)$ is a continuous, strictly decreasing and convex function such that $\phi(1)=0$. The pseudo-inverse $\phi^{[-1]}$ is defined as $$\phi^{[-1]}(t)= \left\{ \begin{array}{l} \phi^{-1}(t), \quad \text{if } 0\leq t \leq \phi(0),\\ 0, \quad \text{if } \phi(0)<t. \end{array} \right.$$

My question is how “small” $\mathcal{C}_0$ is in larger $\mathcal{C}$. I am not sure what meaning and what notion of smallness would be appropriate here.

• I am not sure whether mentioning this is useful but the Fréchet–Hoeffding upper bound $M(u,v) = min(u,v)$ is not Archimedean even though it is exchangeable. We can show though that the Clayton copula with $\theta \rightarrow \infty$ will converge to $M(u,v)$. Commented Sep 11, 2015 at 14:25