Some results on the convergence of (quasi-) copulas

Authors:
Wolfgang Trutschnig
Affiliations:
European Centre for Soft Computing, Edificio Científico Tecnológico, Calle Gonzalo Gutiérrez Quirós, s/n, 3a planta, 33600 Mieres (Asturias), Spain
Venue:
Fuzzy Sets and Systems
Year:
2012

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Abstract

It is shown that pointwise convergence of a sequence (A"n)"n"@?"N of copulas to a copula A is equivalent (1) to the convergence of the corresponding endographs and (2) to the convergence of the corresponding upper (or lower) @a-levels for all but at most countably many @a in [0,1] (all with respect to the Hausdorff metric). Examples are given that show that the countably many exceptions in (2) cannot be omitted. It is furthermore shown that the main results also hold on the bigger class of quasi-copulas.