Wlodzimierz Wysocki

Properties of Diagonal Sections of m-dimensional Archimedean Copulas

896

Abstract

In the paper we discuss some problems related to restrictions of $m$-dimensional copulas to the main diagonal of a $m$-dimensional cube $[0,1]^{m}$ (diagonal sections). We prove that if the diagonal section of $m$-archimedean copula is given, it is easy to get its additive generator by taking a suitable limit. However, this method is applicable to unbounded generators only. A general method is proposed which allows to find unbounded as well as bounded generators. The procedure boils down to finding of a pseudoinverse $y= h(x)$ of an additive generator of $m$-archimedean copula via solving a differential equation $\d{}{y}{x} = {v(y) \over x}$, where $v(y)$ is a certain vector field determined by the diagonal section.
The main results show that almost unifom limits of certain sequences of functions related to absolutely monotone functions of order $m+1$ lead to additve generators of archimedean copulas. Moreover, a new characterization of $m$-archimedean copula is established in terms of limits for iterated diagonal sections. This leads to a concept of a diagonal generator of $m$-archimedean copula.
It is proved that besides a traditional representation $m$-archimedean copula has another representation based on limits of iterated diagonal sections. A class of $m$-copulas having the discussed representation is also investigated.

Key words: $m$-monotonic function, $m$-absolutely monotonic function, diagonal section, $m$-archimedean copula, diagonal generator, vector field, one-parameter group of transformations, generator of one-parameter group.