We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with d neighbors in a germ‐grain model, and the number of degree‐d vertices in an Erdős‐Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.

中文翻译：

---

占用模型的局部极限定理

对于因变量随机和，我们提出了一种相当通用的方法来证明局部极限定理，并且收敛速度很好。当可以显示Stein耦合时，该方法适用。我们的方法涉及斯坦因分布近似方法和斯坦因集中方法。作为应用，证明具有收敛的速度与细菌的数量局部中心极限定理 d的胚芽粒模型的邻居，和功名的数量d的顶点在鄂尔多斯-莱利随机图。在这两种情况下，错误率都是最佳的，最高可达对数因子。