Berry-Esséen Theorem

If  is a probability distribution with zero mean and

(1)

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

(2)

where  is the normal distribution function,  in Feller's notation, and

(3)

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

REFERENCES:

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.