  We study the linear map sending the numerator of the rational function
representing the Hilbert series of a module to that of its r-th Veronese
submodule. We show that the asymptotic behaviour as r tends to infinity depends
on the multidegree of the module and the underlying positively multigraded
polynomial ring. More importantly, we give a polyhedral description for the
asymptotic polynomial and prove that the coefficients are log-concave. In
particular, we extend some results by Beck-Stapledon and Diaconis-Fulman beyond
the standard graded situation.
