论文标题
多项式分布和某些应用的精确局部限制定理
A precise local limit theorem for the multinomial distribution and some applications
论文作者
论文摘要
在Siotani＆Fujikoshi（1984）中，通过反转傅立叶变换来得出多项式分布的精确局部限制定理，在该傅立叶变换中，该错误项已明确至$ n^{ - 1} $。在本文中，我们根据Stirling的公式和仔细处理Taylor扩展提供了替代性（概念上的简单）证明，并展示了结果如何用于近似于$ \ Mathbb {r}^d $的大多数子集的多项式概率。此外，我们讨论了结果的最新应用，以获得伯恩斯坦估计器在单纯形上的渐近特性，我们改善了Carter（2002）对多官方和多元正常实验之间的LE CAM距离结合的主要结果，同时简化了证明，并提及了与精心调节的连续性连续性校正有关的另一种潜在应用。
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.
