We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Amp\`ere equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in Kantorovitch-Wasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension.

翻译：本文描述了利用矩映射构造Stein核的方法，该核是Monge-Ampère方程变体的解。作为推论，我们展示了这些映射的正则性界如何控制经典中心极限定理中的收敛速度，并在对数凹情形下推导出Kantorovitch-Wasserstein距离中新的收敛速率，其中维度依赖关系呈现显式的多项式形式。