Central limit theorem for toric Kähler manifolds

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, \omega, L, h)$ are sequences measures $\{\mu_k^z\}_{k=1}^{\infty}$ parametrized by points $z \in M$. For each $z$ in open orbit, we prove central limit theorem for $\mu_k^z$. The center mass $\mu_k^z$ is image under moment map; after re-centering at $0$ and dilating $\sqrt{k}$, re-normalized measure tends centered Gaussian whose variance Hessian potential $z$. We further give remainder estimate Berry-Esseen type. sequence $\{\mu_k^z\}$ generally not convolution powers proofs only involve analysis.