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It is shown that to every Archimedean copula H there corresponds a one-parameter semigroup of transformations of the interval [0,1]. If the elements of the semigroup are diffeomorphisms, then it determines a special function $v_H$ called the vector generator. Its knowledge permits finding a pseudoinverse y = h(x) of the additive generator of the Archimedean copula H by solving the differential equation $dy/dx=v_H\left(y\right)/x$ with initial condition $(dh/dx)\left(0\right)=-1$. Weak convergence of Archimedean copulas is characterized in terms of vector...

We construct two pairs $({𝒜}_F^{\left[1\right]},{𝒜}_F^{\left[2\right]})$ and $({𝒜}_{\psi }^{\left[1\right]},{𝒜}_{\psi }^{\left[2\right]})$ of ordered parametric families of symmetric dependence functions. The families of the first pair are indexed by regular distribution functions $F$, and those of the second pair by elements $\psi$ of a specific function family $\psi$. We also show that all solutions of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}u}=\frac{\alpha \left(u\right)}{u(1-u)}y$ for $\alpha$ in a certain function family ${\alpha }_{\mathrm{s}}$ are symmetric dependence functions.

We present three characterizations of $n$-dimensional Archimedean copulas: algebraic, differential and diagonal. The first is due to Jouini and Clemen. We formulate it in a more general form, in terms of an $n$-variable operation derived from a binary operation. The second characterization is in terms of first order partial derivatives of the copula. The last characterization uses diagonal generators, which are “regular” diagonal sections of copulas, enabling one to recover the copulas by means of an...