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