  We prove that the sequence of cones of metric measure spaces converges if the
sequence of base spaces converges in Gromov's box, concentration, and weak
topologies. As an application, we show that the generalized Cauchy distribution
with suitable scaling converges to a half line in the concentration topology as
the dimension diverges to infinity. This is a new example distinguished from
previously known examples such as Gaussian distributions and typical closed
Riemannian manifolds with constant Ricci curvature.
