Berry-Esseen inequality

From Encyclopedia of Mathematics
Revision as of 17:19, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

An inequality giving an estimate of the deviation of the distribution function of a sum of independent random variables from the normal distribution function. Let be independent random variables with the same distribution such that



then, for any ,

where is an absolute positive constant. This result was obtained by A.C. Berry [1] and, independently, by C.G. Esseen [2].


[1] A.C. Berry, "The accuracy of the Gaussian approximation to the sum of independent variables" Trans. Amer. Math. Soc. , 49 (1941) pp. 122–136
[2] C.G. Esseen, "On the Liapunoff limit of error in the theory of probability" Ark. Mat. Astr. Fysik , 28A : 2 (1942) pp. 1–19
[3] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)


The constant can be taken to be , cf. [a1], p. 515 ff.


[a1] W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1966) pp. 210
How to Cite This Entry:
Berry-Esseen inequality. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article