# Berry-Esseen inequality

From Encyclopedia of Mathematics

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

This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article