Berry-Esseen inequality

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