Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric binomial distribution, serving as a natural alternative to the normal distribution in discrete settings. The bounds are given with respect to the total variation and a local limit metric. Under appropriate smoothness properties of the summands, the same order of accuracy as in the Berry-Essen Theorem is achieved. The approximation of the total number of points of a point processes is also considered. The results are applied to the exceedances of the $r$-scans process and to the Mat'ern hardcore point process type I to obtain explicit bounds with respect to the two metrics.
"Symmetric and centered binomial approximation of sums of locally dependent random variables." Electron. J. Probab. 13 756 - 776, 2008. https://doi.org/10.1214/EJP.v13-503