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| In [[probability theory]], the [[central limit theorem]] states that, under certain circumstances, the [[probability distribution]] of the scaled [[sample mean|mean of a random sample]] [[convergence in distribution|converges]] to a [[normal distribution]] as the sample size increases to infinity. Under stronger assumptions, the '''Berry–Esseen theorem''', or '''Berry–Esseen inequality''', gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of [[approximation theory|approximation]] between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured by the [[Kolmogorov–Smirnov distance]]. In the case of [[Independence (probability theory)|independent samples]], the convergence rate is {{math|''n''<sup>−1/2</sup>}}, where {{math|''n''}} is the sample size, and the constant is estimated in terms of the [[Skewness|third]] absolute [[Moment (mathematics)|normalized moments]].
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| ==Statement of the theorem==
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| Statements of the theorem vary, as it was independently discovered by two [[mathematician]]s, [[Andrew C. Berry]] (in 1941) and [[Carl-Gustav Esseen]] (1942), who then, along with other authors, refined it repeatedly over subsequent decades.
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| ===Identically distributed summands===
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| One version, sacrificing generality somewhat for the sake of clarity, is the following:
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| :There exists a positive [[Constant (mathematics)|constant]] ''C'' such that if ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., are [[Independent and identically-distributed random variables|i.i.d. random variables]] with E(''X''<sub>1</sub>) = 0, E(''X''<sub>1</sub><sup>2</sup>) = σ<sup>2</sup> > 0, and E(|''X''<sub>1</sub>|<sup>3</sup>) = ρ < ∞, and if we define
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| ::<math>Y_n = {X_1 + X_2 + \cdots + X_n \over n}</math>
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| :the [[sample mean]], with ''F''<sub>''n''</sub> the [[cumulative distribution function]] of
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| ::<math>{Y_n \sqrt{n} \over {\sigma}},</math><!-- please do not change this formula unless you have read and understood the relevant comments on the talk page -->
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| :and Φ the cumulative distribution function of the [[standard normal distribution]], then for all ''x'' and ''n'',
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| ::<math>\left|F_n(x) - \Phi(x)\right| \le {C \rho \over \sigma^3\,\sqrt{n}}.\ \ \ \ (1)</math>
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| [[Image:BerryEsseenTheoremCDFGraphExample.png|thumb|250px|Illustration of the difference in cumulative distribution functions alluded to in the theorem.]]
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| That is: given a sequence of [[independent and identically-distributed random variables]], each having [[mean]] zero and positive [[variance]], if additionally the third absolute [[moment (mathematics)|moment]] is finite, then the [[cumulative distribution function]]s of the [[Standard score|standardized]] sample mean and the standard normal distribution differ (vertically, on a graph) by no more than the specified amount. Note that the approximation error for all ''n'' (and hence the limiting rate of convergence for indefinite ''n'' sufficiently large) is bounded by the [[Big O notation|order]] of ''n''<sup>−1/2</sup>.
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| Calculated values of the constant ''C'' have decreased markedly over the years, from the original value of 7.59 by {{harvtxt|Esseen|1942}}, to 0.7882 by {{harvtxt|van Beek|1972}}, then 0.7655 by {{harvtxt|Shiganov|1986}}, then 0.7056 by {{harvtxt|Shevtsova|2007}}, then 0.7005 by {{harvtxt|Shevtsova|2008}}, then 0.5894 by {{harvtxt|Tyurin|2009}}, then 0.5129 by {{harvtxt|Korolev|Shevtsova|2009}}, then 0.4785 by {{harvtxt|Tyurin|2010}}. The detailed review can be found in the papers {{harvtxt|Korolev|Shevtsova|2009}}, {{harvtxt|Korolev|Shevtsova|2010}}. The best estimate {{Asof|2012|lc=yes}}, ''C'' < 0.4748, follows from the inequality
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| :<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le {0.33554 (\rho+0.415\sigma^3)\over \sigma^3\,\sqrt{n}},</math>
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| due to {{harvtxt|Shevtsova|2011}}, since σ<sup>3</sup> ≤ ρ and 0.33554 · 1.415 < 0.4748. However, if ρ ≥ 1.286σ<sup>3</sup>, then the estimate
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| :<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le {0.3328 (\rho+0.429\sigma^3)\over \sigma^3\,\sqrt{n}},</math>
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| which is also proved in {{harvtxt|Shevtsova|2011}}, gives an even tighter upper estimate. | |
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| {{harvtxt|Esseen|1956}} proved that the constant also satisfies the lower bound
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| : <math>
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| C\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}} \approx 0.40973 \approx \frac{1}{\sqrt{2\pi}} + 0.01079 .
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| </math>
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| ===Non-identically distributed summands===
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| :Let ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., be independent random variables with [[expected value|E]](''X''<sub>i</sub>) = 0, E(''X''<sub>i</sub><sup>2</sup>) = σ<sub>i</sub><sup>2</sup> > 0, and E(|''X''<sub>i</sub>|<sup>3</sup>) = ρ<sub>i</sub> < ∞. Also, let | |
| ::<math>S_n = {X_1 + X_2 + \cdots + X_n \over \sqrt{\sigma_1^2+\sigma_2^2+\cdots+\sigma_n^2} }</math>
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| :be the normalized ''n''-th partial sum. Denote ''F''<sub>''n''</sub> the [[cumulative distribution function|cdf]] of ''S''<sub>''n''</sub>, and Φ the cdf of the [[standard normal distribution]]. For the sake of convenience denote
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| ::<math>\vec{\sigma}=(\sigma_1,...,\sigma_n),\ \vec{\rho}=(\rho_1,...,\rho_n).</math>
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| :In 1941, [[Andrew C. Berry]] proved that for all ''n'' there exists an absolute constant ''C''<sub>1</sub> such that
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| ::<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le C_1\cdot\psi_1,\ \ \ \ (2)</math>
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| :where
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| ::<math>\psi_1=\psi_1\big(\vec{\sigma},\vec{\rho}\big)=\Big({\textstyle\sum\limits_{i=1}^n\sigma_i^2}\Big)^{-1/2}\cdot\max_{1\le
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| i\le n}\frac{\rho_i}{\sigma_i^2}.</math>
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| :Independently, in 1942, [[Carl-Gustav Esseen]] proved that for all ''n'' there exists an absolute constant ''C''<sub>0</sub> such that
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| ::<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le C_0\cdot\psi_0, \ \ \ \ (3)</math>
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| :where
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| ::<math>\psi_0=\psi_0\big(\vec{\sigma},\vec{\rho}\big)=\Big({\textstyle\sum\limits_{i=1}^n\sigma_i^2}\Big)^{-3/2}\cdot\sum\limits_{i=1}^n\rho_i.</math>
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| It is easy to make sure that ψ<sub>0</sub>≤ψ<sub>1</sub>. Due to this circumstance inequality (3) is conventionally called the Berry–Esseen inequality, and the quantity ψ<sub>0</sub> is called the Lyapunov fraction of the third order. Moreover, in the case where the summands ''X''<sub>1</sub>,... ''X''<sub>n</sub> have identical distributions
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| ::<math>\psi_0=\psi_1=\frac{\rho_1}{\sigma_1^3\sqrt{n}},</math> | |
| and thus the bounds stated by inequalities (1), (2) and (3) coincide.
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| Regarding ''C''<sub>0</sub>, obviously, the lower bound established by {{harvtxt|Esseen|1956}} remains valid:
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| : <math>
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| C_0\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}} = 0.4097\ldots.
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| </math>
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| The upper bounds for ''C''<sub>0</sub> were subsequently lowered from the original estimate 7.59 due to {{harvtxt|Esseen|1942}} to (we mention the recent results only) 0.9051 due to {{harvtxt|Zolotarev|1967}}, 0.7975 due to {{harvtxt|van Beek|1972}}, 0.7915 due to {{harvtxt|Shiganov|1986}}, 0.6379 and 0.5606 due to {{harvtxt|Tyurin|2009}} and {{harvtxt|Tyurin|2010}}. {{Asof|2011}} the best estimate is 0.5600 obtained by {{harvtxt|Shevtsova|2010}}.
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| ==See also==
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| *[[Chernoff's inequality]]
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| *[[Edgeworth series]]
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| *[[List of inequalities]]
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| *[[List of mathematical theorems]]
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| ==References==
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| {{refbegin}}
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| * {{cite journal| first=Andrew C. | last=Berry | year=1941
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| | title=The Accuracy of the Gaussian Approximation to the Sum of Independent Variates
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| | journal=Transactions of the American Mathematical Society
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| | volume=49 | issue=1 | pages=122–136 | ref=harv| jstor=1990053| doi=10.1090/S0002-9947-1941-0003498-3
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| }}
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| * Durrett, Richard (1991). ''Probability: Theory and Examples''. Pacific Grove, CA: Wadsworth & Brooks/Cole. ISBN 0-534-13206-5.
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| * {{cite journal | ref=harv
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| | last = Esseen | first = Carl-Gustav
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| | title = On the Liapunoff limit of error in the theory of probability
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| | year = 1942
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| | journal = Arkiv för matematik, astronomi och fysik | issn = 0365-4133
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| | volume = A28
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| | pages = 1–19
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| }}
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| * {{cite journal | ref=harv
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| | last = Esseen | first = Carl-Gustav
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| | title = A moment inequality with an application to the central limit theorem
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| | year = 1956
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| | journal = Skand. Aktuarietidskr.
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| | volume = 39
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| | pages = 160–170
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| }}
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| * Feller, William (1972). ''An Introduction to Probability Theory and Its Applications, Volume II'' (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-25709-5.
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| * {{cite journal | ref=harv
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| | last1 = Korolev | first1 = V. Yu. | last2 = Shevtsova | first2 = I. G.
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| | title = On the upper bound for the absolute constant in the Berry–Esseen inequality
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| | year = 2010
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| | journal = Theory of Probability and its Applications
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| | volume = 54 | issue = 4
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| | pages = 638–658
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| | doi = 10.1137/S0040585X97984449
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| }}
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| * {{cite journal | ref=harv
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| | last1 = Korolev | first1 = Victor | last2 = Shevtsova | first2 = Irina
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| | title = An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums
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| | year = 2010
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| | journal = Scandinavian Actuarial Journal
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| | doi= 10.1080/03461238.2010.485370 | pages=1–25
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| | url= http://www.tandfonline.com/doi/abs/10.1080/03461238.2010.485370
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| }}
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| * Manoukian, Edward B. (1986). ''Modern Concepts and Theorems of Mathematical Statistics''. New York: Springer-Verlag. ISBN 0-387-96186-0.
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| * Serfling, Robert J. (1980). ''Approximation Theorems of Mathematical Statistics''. New York: John Wiley & Sons. ISBN 0-471-02403-1.
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| * {{cite journal | ref=harv
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| | last = Shevtsova | first = I. G.
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| | title = On the absolute constant in the Berry–Esseen inequality
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| | year = 2008
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| | journal = The Collection of Papers of Young Scientists of the Faculty of Computational Mathematics and Cybernetics
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| | issue = 5
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| | pages = 101–110
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| }}
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| * {{cite journal | ref=harv
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| | last = Shevtsova | first = I. G.
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| | title = Sharpening of the upper bound of the absolute constant in the Berry–Esseen inequality
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| | year = 2007
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| | journal = Theory of Probability and its Applications
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| | volume = 51 | issue = 3
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| | pages = 549–553
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| | doi= 10.1137/S0040585X97982591
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| }}
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| * {{cite journal | ref=harv
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| | last = Shevtsova | first = I. G.
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| | title = An Improvement of Convergence Rate Estimates in the Lyapunov Theorem
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| | year = 2010
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| | journal = Doklady Mathematics
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| | volume = 82 | issue = 3
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| | pages = 862–864
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| | doi= 10.1134/S1064562410060062
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| }}
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| * {{cite arXiv | ref=harv
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| | last = Shevtsova | first = Irina
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| | title = On the absolute constants in the Berry Esseen type inequalities for identically distributed summands
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| | year = 2011
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| | eprint = 1111.6554
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| | class = math.PR
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| }}
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| * {{cite journal | ref=harv
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| | last = Shiganov | first = I.S.
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| | title = Refinement of the upper bound of a constant in the remainder term of the central limit theorem
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| | year = 1986
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| | journal = Journal of Soviet mathematics
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| | volume = 35
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| | pages = 109–115
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| | doi=10.1007/BF01121471 | issue=3
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| }}
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| * {{cite journal | ref=harv
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| | last = Tyurin | first = I.S.
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| | title = On the accuracy of the Gaussian approximation
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| | year = 2009
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| | journal = Doklady Mathematics
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| | volume = 80 | issue = 3
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| | pages = 840–843 | doi=10.1134/S1064562409060155
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| }}
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| * {{cite journal | ref=harv
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| | last = Tyurin | first = I.S.
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| | title = An improvement of upper estimates of the constants in the Lyapunov theorem
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| | year = 2010
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| | journal = Russian Mathematical Surveys
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| | volume = 65 | issue = 3(393)
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| | pages = 201–202
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| }}
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| * {{cite journal | ref=harv
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| | last = van Beek | first = P.
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| | title = An application of Fourier methods to the problem of sharpening the Berry–Esseen inequality
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| | year = 1972
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| | journal = Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete
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| | volume = 23
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| | pages = 187–196
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| | doi=10.1007/BF00536558 | issue=3
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| }}
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| * {{cite journal | ref=harv
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| | last = Zolotarev | first = V. M.
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| | title = A sharpening of the inequality of Berry–Esseen
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| | year = 1967
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| | journal = Z. Wahrsch. Verw. Geb.
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| | volume = 8
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| | pages = 332–342 | doi=10.1007/BF00531598 | issue=4
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| }}
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| {{refend}}
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| ==External links==
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| * Gut, Allan & Holst Lars. [http://www.stat.unipd.it/bernoulli/02a/bn_3.html Carl-Gustav Esseen], retrieved Mar. 15, 2004.
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| * {{springer|title=Berry-Esseen inequality|id=p/b015760}}
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| {{DEFAULTSORT:Berry-Esseen theorem}}
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| [[Category:Probability theorems]]
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| [[Category:Probabilistic inequalities]]
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| [[Category:Statistical theorems]]
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| [[Category:Central limit theorem]]
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