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Włodzimierz Wysocki
Induced semigroup of an Archimedean copula
960
Abstract
It is shown that with every Archimedean copula $H$ there is associated
a one-parameter semigroup of transformations of the interval
$[0,1]$.
If the elements of the semigroup are diffeomorphisms,
then the semigroup determines a special function vH
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 $\dd{}{y}{x} = {v_{H}(y) \over x}$
with initial condition $\dd{}{h}{x}(0) = -1$.
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 the projection of a copula.
Keywords :
Phase flow, generator of phase flow, Archimedean copula,
vector generator of Archimedean copula, projection of copula
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