In Siotani and Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order $N^{-1}$. In this paper, we give an alternative (conceptually simpler) proof based on Stirling’s formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of ${\mathbb{R}}^d$. Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter (2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.

在Siotani和Fujikoshi（1984）中，多项式分布的精确局部极限定理是通过对傅立叶变换求逆而得出的，其中误差项是显式的。 ${ñ}^{-\mathrm{1个}}$。在本文中，我们根据斯特林公式和对泰勒展开的谨慎处理给出了另一种（概念上更简单）的证明，并且我们展示了如何将结果用于近似大多数子集的多项式概率${\mathbb{[R}}^d$。此外，我们讨论了该结果最近用于获得单纯形上Bernstein估计的渐近性质的应用，我们改进了Carter（2002）关于多项式和多元正态实验之间的Le Cam距离界限的主要结果，同时简化了证明，并且我们提到了另一个与微调的连续性校正有关的潜在应用。