如果 
是一个均值为零的概率分布，且
 |
(1)
|
其中上述积分是斯蒂尔吉斯积分，则对于所有 
和 
,
 |
(2)
|
其中 
是正态分布函数，
在 Feller 的记号中，且
 |
(3)
|
是 
-重 卷积 
的归一化 (Wallace 1958, Feller 1971)。
Berry-Esséen 定理
另请参阅
中心极限定理使用 Wolfram|Alpha 探索
参考文献
Bergström, H. "On the Central Limit Theorem." Skand. Aktuarietidskr. 27, 139-153, 1944.Bergström, H. "On the Central Limit Theorem in the Space
, 
." Skand. Aktuarietidskr. 28, 106-127, 1945.Bergström, H. "On the Central Limit Theorem in the Case of not Equally Distributed Random Variables." Skand. Aktuarietidskr. 32, 37-62, 1949.Berry, A. C. "The Accuracy of the Gaussian Approximation to the Sum of Independent Variates." Trans. Amer. Math. Soc. 49, 122-136 1941.Esseen, C. G. "On the Liapounoff Limit of Error in the Theory of Probability." Ark. Mat. Astr. och Fys. 28A, No. 9, 1-19, 1942.Esseen, C. G. "Fourier Analysis of Distribution Functions." Acta Math. 77, 1-125, 1945.Esseen, C. G. "A Moment Inequality with an Application to the Central Limit Theorem." Skand. Aktuarietidskr. 39, 160-170, 1956.Feller, W. "The Berry-Esséen Theorem." §16.5 in An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 542-546, 1971.Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, p. 369, 1988.Hsu, P. L. "The Approximate Distribution of the Mean and Variance of a Sample of Independent Variables." Ann. Math. Stat. 16, 1-29, 1945.Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.在 Wolfram|Alpha 中被引用
Berry-Esséen 定理请引用为
Weisstein, Eric W. "Berry-Esséen 定理。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/Berry-EsseenTheorem.html