标题： 无效正态近似的精度界限 显示英文标题

标题： Bounds for the accuracy of invalid normal approximation

摘要： 在应用概率中，通常对具有假设加法结构的数据分布使用正态近似。这种传统基于独立随机变量和的中心极限定理。然而，当观察到的样本量有限时，实际上不可能检查确保中心极限定理有效性的条件。因此，了解在理论上不适用的情况下使用正态近似的真实准确性非常重要。此外，在与计算机模拟相关的某些情况下，如果求和中各个项的分布属于特征指数小于二的稳定律的吸引域，则观测到的归一化和分布与正态分布之间的距离随着项数的增加而先减小，只有当项数足够大时才开始增加。本文试图对此现象进行一些理论解释。 显示英文摘要

摘要： In applied probability, the normal approximation is often used for the distribution of data with assumed additive structure. This tradition is based on the central limit theorem for sums of (independent) random variables. However, it is practically impossible to check the conditions providing the validity of the central limit theorem when the observed sample size is limited. Therefore it is very important to know what the real accuracy of the normal approximation is in the cases where it is used despite it is theoretically inapplicable. Moreover, in some situations related with computer simulation, if the distributions of separate summands in the sum belong to the domain of attraction of a stable law with characteristic exponent less than two, then the observed distance between the distribution of the normalized sum and the normal law first decreases as the number of summands grows and begins to increase only when the number of summands becomes large enough. In the present paper an attempt is undertaken to give some theoretical explanation to this effect.