Thirty years of copulas. (English) Zbl 0727.60001

Schweizer, Berthold

Advances in probability distributions with given marginals. Beyond the copulas, Lect. Symp., Rome/Italy, Math. Appl. 67, 13-50 (1991).

[For the entire collection see Zbl 0722.00031.]
The author gives a personal review on developments on the notion of copulas in the last thirty years. Copulas are in a more modern terminology distribution functions with given uniform marginals. A basic result of A. Sklar (1959) gives a representation of any distribution function in the form \(C(F_ 1,...,F_ n)\), where C is its copula (uniform representation) and \(F_ i\) are the marginals. The connection with the development of probabilistic metric spaces is described. These relations are applied to the construction of (local) bounds for the distribution of functionals in the class of distributions with given marginals. A further application is to the construction of dependence measures.

Reviewer: L.Rüschendorf (Münster)

MSC:

60A05	Axioms; other general questions in probability
60E05	Probability distributions: general theory
60-03	History of probability theory
01A65	Development of contemporary mathematics

Keywords:

Archimedean t-norm; \(\sigma \) -operation; copulas; probabilistic metric spaces; dependence measures

Citations:

Zbl 0722.00031