Properties of the induced semigroup of an Archimedean copula
Volume 31 / 2004
Applicationes Mathematicae 31 (2004), 161-174
MSC: Primary 62H05, 60E05.
DOI: 10.4064/am31-2-3
Abstract
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 ${d^{}{y}/d{x}^{}} = {v_{H}(y) / x}$ with initial condition ${(d^{}{h}/d{x}^{})}(0) = -1$. Weak convergence of Archimedean copulas is characterized in terms of vector generators. A new characterization of Archimedean copulas is also given by using the notion of a projection of a copula.
