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Copulas constructed from diagonal sections. (English) Zbl 0906.60022
Beneš, Viktor (ed.) et al., Distributions with given marginals and moment problems. Proceedings of the 1996 conference, Prague, Czech Republic. Dordrecht: Kluwer Academic Publishers. 129-136 (1997).
Summary: If $$C$$ is a copula, then its diagonal section is the function $$\delta$$ given by $$\delta(t) =C(t,t)$$. It follows that (i) $$\delta (1)=1$$; (ii) $$0\leq \delta (t_2)- \delta (t_1)\leq 2(t_2- t_1)$$ for all $$t_1$$, $$t_2$$ in $$[0,1]$$ with $$t_1\leq t_2$$; and (iii) $$\delta (t)\leq 1$$ for all $$t$$ in $$[0,1]$$. If $$\delta$$ is any function satisfying (i)–(iii), does there exist a copula $$C$$ whose diagonal section is $$\delta$$? We answer this question affirmatively by constructing copulas we call diagonal copulas: $$K(u,v)= \min (u,v,(1/2) [\delta (u)+ \delta(v)])$$. We also present examples, investigate some dependence properties of random variables which have diagonal copulas, and answer an open question about tail dependence in bivariate copulas by H. Joe [J. Multivariate Anal. 46, No. 2, 262-282 (1993; Zbl 0778.62045)].
For the entire collection see [Zbl 0885.00054].

##### MSC:
 6e+100 Distribution theory