A conjecture of Bertino: from copulas to graph theory

Abstract

In this article we prove a conjecture of Salvatore Bertino which centers on a study of the function

F(C) = \int_0^{1/2} V_C([t,t+1/2]^2) dt,

where C is a copula. This function, which arises in the study of the index of dissimilarity between two random variables (see Bertino 1977), is the integral of the C-volumes of the family of squares of sidelength 1/2 sliding from the lower left to the upper right along the ascending diagonal of the unit square. The conjecture made in 1999 is that the minimum value of F among the Bertino copulas is 1/8, which is the value of F at the Fr´echet lower bound. The conjecture is established by showing that a certain family of shuffles of M is dense in the set of Bertino copulas and then solving a problem in weighted graphs, which is equivalent to the conjecture in the restricted case.