Kybernetika 48 no. 2, 287-293, 2012

On a problem by Schweizer and Sklar

Fabrizio Durante


We give a representation of the class of all $n$-dimensional copulas such that, for a fixed $m\in \mathbb N$, $2\le m<n$, all their $m$-dimensional margins are equal to the independence copula. Such an investigation originated from an open problem posed by Schweizer and Sklar.


copulas, distributions with given marginals, Fréchet-Hoeffding bounds, partial mutual independence


60E05, 62E10


  1. V. Beneš, J. Štěpán and eds.: Distributions With Given Marginals and Moment Problems. Kluwer Academic Publishers, Dordrecht 1997.   CrossRef
  2. C. M. Cuadras, J. Fortiana, J. A. Rodriguez-Lallena and eds.: Distributions With Given Marginals and Statistical Modelling. Kluwer Academic Publishers, Dordrecht 2002. Papers from the meeting held in Barcelona 2000.   CrossRef
  3. G. Dall'Aglio, S. Kotz, G. Salinetti and eds.: Advances in Probability Distributions with Given Marginals. Mathematics and its Applications 67, Kluwer Academic Publishers Group, Dordrecht 1991. Beyond the Copulas, Papers from the Symposium on Distributions with Given Marginals held in Rome 1990.   CrossRef
  4. P. Deheuvels: Indépendance multivariée partielle et inégalités de Fréchet. In: Studies in Probability and Related Topics, Nagard, Rome 1983, pp. 145-155.   CrossRef
  5. F. Durante, E. P. Klement and J. J. Quesada-Molina: Bounds for trivariate copulas with given bivariate marginals. J. Inequal. Appl. 2008 (2008), 1-9.   CrossRef
  6. P. Jaworski, F. Durante, W. Härdle, T. Rychlik and eds.: Copula Theory and its Applications. Lecture Notes in Statistics - Proceedings 198, Springer, Berlin - Heidelberg 2010.   CrossRef
  7. H. Joe: Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73, Chapman \& Hall, London 1997.   CrossRef
  8. R. B. Nelsen: An Introduction to Copulas. Second edition. Springer Series in Statistics, Springer, New York 2006.   CrossRef
  9. L. Rüschendorf, B. Schweizer, M. D. Taylor and eds.: Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics, Lecture Notes - Monograph Series 28, Hayward 1996.   CrossRef
  10. B. Schweizer and A. Sklar: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing Co., New York 1983. Reprinted, Dover, Mineola 2005.   CrossRef
  11. K. F. Siburg and P. A. Stoimenov: Gluing copulas. Comm. Statist. Theory Methods 37 (2008), 19, 3124-3134.   CrossRef
  12. A. Sklar: Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229-231.   CrossRef