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Berry-Esséen Theorem


If F(x) is a probability distribution with zero mean and

 rho=int_(-infty)^infty|x|^3dF(x)<infty,
(1)

where the above integral is a stieltjes integral, then for all x and n,

 |F_n(x)-Phi(x)-1/2|<(33)/4rho/(sigma^3sqrt(n)),
(2)

where Phi(x) is the normal distribution function, Phi(x)+1/2=N(x) in Feller's notation, and

 F_n(x)=F^(n*)(xsigmasqrt(n))
(3)

is the normalized n-fold convolution of F(x) (Wallace 1958, Feller 1971).


See also

Central Limit Theorem

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References

Bergström, H. "On the Central Limit Theorem." Skand. Aktuarietidskr. 27, 139-153, 1944.Bergström, H. "On the Central Limit Theorem in the Space R_k, k>1." 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.

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Berry-Esséen Theorem

Cite this as:

Weisstein, Eric W. "Berry-Esséen Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Berry-EsseenTheorem.html

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