Constructing archimedean copulas from diagonal sections

We introduce a family ℱℱ of functions called diagonal generators. These are convex functions with the properties of diagonal sections of archimedean copulas. We show that to each diagonal generator ff there corresponds an archimedean copula CfCf with the asymptotic representation Cf(u1,u2)=limk→∞fk[f−k(u1)+f−k(u2)−1]Cf(u1,u2)=limk→∞fk[f−k(u1)+f−k(u2)−1]. Moreover, the diagonal section of CfCf equals ff.We characterize archimedean copulas in terms of their asymptotic form. We construct a family ℱFℱF of diagonal generators, induced by a regular distribution function FF. We study a differential equation (depending on a function parameter), whose solution is FF. We give four applications of diagonal generators: to concordance, quadrant dependence, measures of dependence and convergence of copulas.