Limits of polyhedral multinomial distributions

We consider limits of certain measures supported on lattice points in lattice polyhedra defined as the intersection of half-spaces $\{m\in {\mathbb{R}}^n\mathrm{\mid }⟨v_i,x⟩+a_i\ge 0\}$, where ${\sum }_i{v}_i=0$. The measures are densities associated to lattice random variables obtained by restriction of multinomial random variables. We find the limiting Gaussian distributions explicitly.