In Siotani & 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.
翻译:在Siotani & Fujikoshi(1984年)中,通过反转Fourier变换,得出了多元分配的确切本地限值,其中错误术语的清晰度最高为$N ⁇ -1}美元。在本文中,我们根据Stirling的公式和对泰勒扩张的仔细处理,给出了另一种(概念上更简单的)证据,我们展示了结果如何用于在大部分子集($\mathbb{R ⁇ d$)上大致接近多位概率。此外,我们讨论了最近对结果的应用,以获得Bernstein估测器在简单x上的无症状特性,我们改进了卡特(2002年)关于多位数和多变数正常实验之间距离的主要结果,同时简化了证据,我们提到了另一个与微调的连续性校正相关的潜在应用。