  Associated to the Bergman kernels of a polarized toric Kaehler manifold $(M,
\omega, L, h)$ are sequences of measures $\{\mu_k^z\}_{k=1}^{\infty}$
parametrized by the points $z \in M$. We determine the asymptotics of the
entropies $H(\mu_k^z)$ of these measures. The sequence $\mu_k^z$ in some ways
resembles a sequence of convolution powers; we determine precisely when it
actually is such a sequence.
