Geodesic
Properties of the induced semigroup of an Archimedean copula
Włodzimierz Wysocki1
1 Institute of Foundations of Computer Science Polish Academy of Sciences Ordona 21 01-237 Warszawa, Poland
Applicationes Mathematicae, Tome 31 (2004) no. 2, pp. 161-174 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 ${d^{}{y}/d{x}^{}} = {v_{H}(y) / x}$ 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.
DOI : 10.4064/am31-2-3
Keywords: shown every archimedean copula there corresponds one parameter semigroup transformations interval elements semigroup diffeomorphisms determines special function called vector generator its knowledge permits finding pseudoinverse additive generator archimedean copula solving differential equation initial condition weak convergence archimedean copulas characterized terms vector generators characterization archimedean copulas given using notion projection copula

Włodzimierz Wysocki 1

1 Institute of Foundations of Computer Science Polish Academy of Sciences Ordona 21 01-237 Warszawa, Poland 
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Włodzimierz Wysocki. Properties of the induced semigroup
 of an Archimedean copula. Applicationes Mathematicae, Tome 31 (2004) no. 2, pp. 161-174. doi: 10.4064/am31-2-3

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