Berry-Esseen inequality

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2010 Mathematics Subject Classification: Primary: 60F05 [MSN][ZBL]

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

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


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


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

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]. Feller obtained an explicit value for the constant $A \le 33/4$ , see [Fe, p. 515]: it is now known that $A \le 0.7655$, see [Fi, p. 264].


[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
[Fi] Steven R. Finch, "Mathematical Constants", Cambridge University Press (2003) ISBN 0-521-81805-2 Sect. 4.6
[Pe] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042
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