关于对称超几何律的正规逼近
On normal approximations to symmetric hypergeometric laws
摘要：
计算了具有标准差$σ$的对称超几何律与其通常的正态近似之间的Kolmogorov距离，并证明其小于$1/(\sqrt{8π}\,σ)$，且阶数$1/σ$和常数$1/\sqrt{8π}$是最优的. Hipp和Mattner（2007）关于对称二项式定律的结果是作为特例得到的。连接到Berry-Esseen类型的结果在更一般的情况下，简单的随机样本或伯努利卷积的总和进行了解释。独立感兴趣的辅助结果包括相当尖锐的正态分布函数不等式，超几何律的一个简单的可识别性结果，以及与Lévy的浓度方差不等式有关的一些评论。
英文摘要：
The Kolmogorov distances between a symmetric hypergeometric law with standard deviation $σ$ and its usual normal approximations are computed and shown to be less than $1/(\sqrt{8π}\,σ)$, with the order $1/σ$ and the constant $1/\sqrt{8π}$ being optimal. The results of Hipp and Mattner (2007) for symmetric binomial laws are obtained as special cases. Connections to Berry-Esseen type results in more general situations concerning sums of simple random samples or Bernoulli convolutions are explained. Auxiliary results of independent interest include rather sharp normal distribution function inequalities, a simple identifiability result for hypergeometric laws, and some remarks related to Lévy's concentration-variance inequality.
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