Functional equations involving Sibuya's dependence function (Q1647751)

The authors consider the problem of solving the functional equation \(D_{\mathbf X}(x_1,x_2)=D_{{\mathbf X}_{(t_1,t_2)}}(x_1,x_2)\) for all \(x_1\), \(x_2\), \(t_1\), \(t_2\geq 0\), where \(D_{\mathbf X}(x_1,x_2)\) is the Sibuya's dependence function and \(\mathbf X\) and \({\mathbf X}_{(t_1,t_2)}\) are random vectors defined as \({\mathbf X}=(X_1,X_2)\) and \({\mathbf X}_{(t_1,t_2)}=[(X_1-t_1,X_2-t_2)|X_1>t_1,X_2>t_2]\), respectively. It is shown that the ``non-aging'' dependence function \(D_{\mathbf X}(x_1,x_2)\) can be characterized by an equivalent functional equation based on the corresponding survival functions. Beside this, the authors prove that the general solution of the above functional equation is given by \(D_{\mathbf X}(x_1,x_2)=c x_1x_2\) for some constant \(c\). At the end, a new characterization of Gumbel's bivariate exponential distribution is obtained through the equivalent functional equation based on the corresponding survival functions.