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 $X_1,\ldots,X_n$ be independent random variables with the same distribution such that

$$\mathbf{E}X_j=0$,\quad \mathbf{E}X_j^2=\sigma^2>0,\quad\mathbf{E}\lvert X_j\rvert^3<\infty.$$

Let

$$\rho=\frac{\mathbf{E}\lvert X_j\rvert^3}{\sigma^3}$$

and

$$\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,\mathrm{d}t;$$

then, for any $n$,

$$\sup_x\left\lvert\mathbf{P}\left\{\frac{1}{\sigma\sqrt{n}}\sum_{j=1}^nX_j\leq x\right\}-\Phi(x)\right\rvert\leq A\frac{\rho}{\sqrt{n}},$$

where $A$ is an absolute positive constant. This result was obtained by A.C. Berry [Be] and, independently, by C.G. Esseen [Es]. The constant $A$ can be taken to be $33/4$ , see [Fe, p. 515].

References

[Be]	A.C. Berry, "The accuracy of the Gaussian approximation to the sum of independent variables" Trans. Amer. Math. Soc. , 49 (1941) pp. 122–136
[Es]	C.G. Esseen, "On the Liapunoff limit of error in the theory of probability" Ark. Mat. Astr. Fysik , 28A : 2 (1942) pp. 1–19
[Fe]	W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1966) pp. 210
[Pe]	V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)