Berry-Esseen inequality
From Encyclopedia of Mathematics
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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
 |
Let
 |
and
 |
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].
References
| [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) |
Comments
The constant 
can be taken to be 
, cf. [a1], p. 515 ff.
References
| [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: http://encyclopediaofmath.org/index.php?title=Berry-Esseen_inequality&oldid=16984
Berry-Esseen inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berry-Esseen_inequality&oldid=16984
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article