View source for Berry–Esseen theorem

{{Short description|Theorem in probability theory}}
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 test#Kolmogorov–Smirnov statistic|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 moment]]. It is also possible to give non-uniform bounds which become more strict for more extreme events.

==Statement of the theorem==
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.

===Identically distributed summands===

One version, sacrificing generality somewhat for the sake of clarity, is the following:

: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 [[Expected value|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'' < ∞,<ref group="note">Since the random variables are identically distributed, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... all have the same [[moment (mathematics)|moments]] as ''X''<sub>1</sub>.</ref> and if we define
::<math>Y_n = {X_1 + X_2 + \cdots + X_n \over n}</math>
:the [[sample mean]], with ''F''<sub>''n''</sub> the [[cumulative distribution function]] of
::<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 -->
:and Φ the cumulative distribution function of the [[standard normal distribution]], then for all ''x'' and ''n'',
::<math>\left|F_n(x) - \Phi(x)\right| \le {C \rho \over \sigma^3\sqrt{n}}.\ \ \ \ (1)</math>

[[Image:BerryEsseenTheoremCDFGraphExample.png|thumb|250px|Illustration of the difference in cumulative distribution functions alluded to in the theorem.]]
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>.

Calculated upper bounds on the constant ''C'' have decreased markedly over the years, from the original value of 7.59 by Esseen in 1942.<ref>{{harvtxt|Esseen|1942}}. For improvements see {{harvtxt|van Beek|1972}}, {{harvtxt|Shiganov|1986}}, {{harvtxt|Shevtsova|2007}}, {{harvtxt|Shevtsova|2008}},  {{harvtxt|Tyurin|2009}}, {{harvtxt|Korolev|Shevtsova|2010a}}, {{harvtxt|Tyurin|2010}}. The detailed review can be found in the papers {{harvtxt|Korolev|Shevtsova|2010a}} and {{harvtxt|Korolev|Shevtsova|2010b}}.</ref> The estimate ''C''&nbsp;<&nbsp;0.4748 follows from the inequality
:<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>
since ''σ''<sup>3</sup>&nbsp;≤&nbsp;''ρ'' and 0.33554&nbsp;·&nbsp;1.415&nbsp;<&nbsp;0.4748. However, if ''ρ''&nbsp;≥&nbsp;1.286''σ''<sup>3</sup>, then the estimate 
:<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>
is even tighter.{{sfnp|Shevtsova|2011}}

{{harvtxt|Esseen|1956}} proved that the constant also satisfies the lower bound
: <math>
    C\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}} \approx 0.40973 \approx \frac{1}{\sqrt{2\pi}} + 0.01079 .
  </math>

===Non-identically distributed summands===

: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>
: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  
::<math>\vec{\sigma}=(\sigma_1,\ldots,\sigma_n),\ \vec{\rho}=(\rho_1,\ldots,\rho_n).</math>
:In 1941, [[Andrew C. Berry]] proved that for all ''n'' there exists an absolute constant ''C''<sub>1</sub> such that
::<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le C_1\cdot\psi_1,\ \ \ \ (2)</math>
:where
::<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
i\le n}\frac{\rho_i}{\sigma_i^2}.</math>

:Independently, in 1942, [[Carl-Gustav Esseen]] proved that for all ''n'' there exists an absolute constant ''C''<sub>0</sub> such that
::<math>\sup_{x\in\mathbb R}\left|F_n(x) - \Phi(x)\right| \le C_0\cdot\psi_0, \ \ \ \ (3)</math>
:where
::<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>

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 
::<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 apart from the constant.

Regarding ''C''<sub>0</sub>, obviously, the lower bound established by {{harvtxt|Esseen|1956}} remains valid:
: <math>
    C_0\geq\frac{\sqrt{10}+3}{6\sqrt{2\pi}} = 0.4097\ldots.
  </math>

The lower bound is exactly reached only for certain Bernoulli distributions (see {{harvtxt|Esseen|1956}} for their explicit expressions).

The upper bounds for ''C''<sub>0</sub> were subsequently lowered from Esseen's original estimate 7.59 to 0.5600.<ref>{{harvtxt|Esseen|1942}}; {{harvtxt|Zolotarev|1967}}; {{harvtxt|van Beek|1972}}; {{harvtxt|Shiganov|1986}}; {{harvtxt|Tyurin|2009}}; {{harvtxt|Tyurin|2010}}; {{harvtxt|Shevtsova|2010}}.</ref>

===Sum of a random number of random variables===

Berry–Esseen theorems exist for the sum of a random number of random variables. The following is Theorem 1 from Korolev (1989), substituting in the constants from Remark 3.<ref>{{cite journal |last1=Korolev |first1=V. Yu |title=On the Accuracy of Normal Approximation for the Distributions of Sums of a Random Number of Independent Random Variables |journal=Theory of Probability & Its Applications |date=1989 |volume=33 |issue=3 |pages=540–544 |doi=10.1137/1133079}}</ref> It is only a portion of the results that they established:

:Let <math>\{X_i\}</math> be independent, identically distributed random variables with <math>E(X_i) = \mu</math>, <math>\operatorname{Var}(X_i) = \sigma^2</math>, <math>E|X_i - \mu|^3 = \kappa^3</math>. Let <math>N</math> be a non-negative integer-valued random variable, independent from <math>\{X_i\}</math>. Let <math>S_N = X_1 + \cdots + X_N</math>, and define
::<math>
  \Delta = \sup_{x} \left|
    P\left(
      \frac{S_N - E(S_N)}{\sqrt{\operatorname{Var}(S_N)}}
      \leq
      z
    \right)
    -
    \Phi(z)
  \right|
</math>
:Then
::<math>
  \Delta \leq
  3.8696\frac{\kappa^3}{\sqrt{E(N)}\sigma^3} +
  1.0395\frac{E|N - E(N)|}{E(N)} +
  0.2420\frac{\mu^2 \operatorname{Var}(N)}{\sigma^2 E(N)}
</math>

===Multidimensional version===
As with the [[Central limit theorem#Multidimensional CLT|multidimensional central limit theorem]], there is a multidimensional version of the Berry–Esseen theorem.<ref>Bentkus, Vidmantas. "A Lyapunov-type bound in R<sup>d</sup>." Theory of Probability & Its Applications 49.2 (2005): 311–323.</ref><ref name=":0" />

:Let <math>X_1,\dots,X_n</math> be independent <math>\mathbb R^d</math>-valued random vectors each having mean zero. Write <math>S_n = \sum_{i=1}^n X_i</math> and assume <math>\Sigma_n = \operatorname{Cov}[S_n]</math> is invertible. Let <math>Z_n\sim\operatorname{N}(0,{\Sigma_n})</math> be a <math>d</math>-dimensional Gaussian with the same mean and covariance matrix as <math>S_n</math>. Then for all convex sets <math>U\subseteq\mathbb R^d</math>,
::<math>\big|\Pr[S_n\in U]-\Pr[Z_n\in U]\,\big| \le C d^{1/4} \gamma_n</math>,
:where <math>C</math> is a universal constant and <math>\gamma_n=\sum_{i=1}^n \operatorname{E}\big[\|\Sigma_n^{-1/2}X_i\|_2^3\big]</math> (the third power of the [[L2 norm|L<sup>2</sup> norm]]).

The dependency on <math>d^{1/4}</math> is conjectured to be optimal, but might not be.<ref name=":0">{{Cite journal|last=Raič|first=Martin|date=2019|title=A multivariate Berry--Esseen theorem with explicit constants|journal=Bernoulli|volume=25|issue=4A|pages=2824–2853|doi=10.3150/18-BEJ1072|issn=1350-7265|arxiv=1802.06475|s2cid=119607520}}</ref><!-- did you mean "might not necessarily be"? -->

== Non-uniform bounds ==
The bounds given above consider the maximal difference between the cdf's. They are 'uniform' in that they do not depend on <math>x</math> and quantify the uniform convergence <math>F_n \to \Phi</math>. However, because <math>F_n(x) - \Phi(x)</math> goes to zero for large <math>x</math> by general properties of cdf's, these uniform bounds will be overestimating the difference for such arguments. This is despite the uniform bounds being sharp in general. It is therefore desirable to obtain upper bounds which depend on <math>x</math> and in this way become smaller for large <math>x</math>. 

One such result going back to {{Harvard citation|Esseen|1945}} that was since improved multiple times is the following.

:As above, 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>\sigma^2 = \sum_{i=1}^{n} \sigma_i^2</math> and
::<math>S_n = {X_1 + X_2 + \cdots + X_n \over \sigma}</math>
: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]]. Then 
::<math>|F_n(x) - \Phi(x)| \leq \frac{C_3}{1 + |x|^3} \cdot \sigma^{-3} \sum_{i = 1}^n \rho_i</math>,
:where <math>C_3</math> is a universal constant.

The constant <math>C_3</math> may be taken as 114.667.<ref>{{Cite book |last=Paditz |first=Ludwig |title=Über die Annäherung der Verteilungsfunktionen von Summen unabhängiger Zufallsgrößen gegen unbegrenzt teilbare Verteilungsfunktionen unter besonderer Beachtung der Verteilungsfunktion der standardisierten Normalverteilung |year=1997 |location=Dresden |pages=6 |language=de |trans-title=On the approximation of cumulative distribution functions of sums of independent random variables by infinitely divisible cumulative distribution functions with special attention to the cumulative distribution function of the standard normal distribution}}</ref> Moreover, if the <math>X_i</math> are identically distributed, it can be taken as <math>C + 8(1+\mathrm{e})</math>, where <math>C</math> is the constant from the first theorem above, and hence 30.2211 works.<ref>{{Cite journal |last=Michel |first=R. |date=1981 |title=On the constant in the nonuniform version of the Berry-Esséen theorem |url=https://link.springer.com/article/10.1007/BF01013464 |journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete |volume=55 |pages=109-117}}</ref>

==See also==
*[[Chernoff's inequality]]
*[[Edgeworth series]]
*[[List of inequalities]]
*[[List of mathematical theorems]]
*[[Concentration inequality]]

==Notes==
<references group="note" />

==References==
{{Reflist}}

==Bibliography==
{{refbegin}}
* {{cite journal| first=Andrew C. | last=Berry | year=1941
  | title=The Accuracy of the Gaussian Approximation to the Sum of Independent Variates
  | journal=Transactions of the American Mathematical Society
  | volume=49 | issue=1 | pages=122–136 | jstor=1990053| doi=10.1090/S0002-9947-1941-0003498-3
  | doi-access=free}}
* Durrett, Richard (1991). ''Probability: Theory and Examples''. Pacific Grove, CA: Wadsworth & Brooks/Cole. {{ISBN|0-534-13206-5}}.
* {{cite journal | last = Esseen | first = Carl-Gustav
  | title = On the Liapunoff limit of error in the theory of probability
  | year = 1942
  | journal = Arkiv för Matematik, Astronomi och Fysik | issn = 0365-4133
  | volume = A28
  | pages = 1–19
  }}
* {{cite journal | last = Esseen | first = Carl-Gustav
  | title = Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law
  | year = 1945
  | journal = Acta Mathematica.
  | volume = 77
  | pages = 1–125
  | doi = 10.1007/BF02392223
  }}
* {{cite journal | last = Esseen | first = Carl-Gustav
  | title = A moment inequality with an application to the central limit theorem
  | year = 1956
  | journal = Skand. Aktuarietidskr.
  | volume = 39
  | pages = 160–170
  }}
* 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}}.
* {{cite journal | last1 = Korolev | first1 = V. Yu. | last2 = Shevtsova | first2 = I. G.
  | title = On the upper bound for the absolute constant in the Berry–Esseen inequality
  | year = 2010a
  | journal = Theory of Probability and Its Applications 
  | volume = 54 | issue = 4
  | pages = 638–658
  | doi = 10.1137/S0040585X97984449
  }}
* {{cite journal | last1 = Korolev | first1 = Victor | last2 = Shevtsova | first2 = Irina
  | title = An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums
  | year = 2010b
  | journal = Scandinavian Actuarial Journal
  | volume = 2012 | issue = 2 | doi= 10.1080/03461238.2010.485370 | pages=1–25
  | arxiv= 0912.2795| s2cid = 115164568 }}
* Manoukian, Edward B. (1986). ''Modern Concepts and Theorems of Mathematical Statistics''. New York: Springer-Verlag. {{ISBN|0-387-96186-0}}.
* Serfling, Robert J. (1980). ''Approximation Theorems of Mathematical Statistics''. New York: John Wiley & Sons. {{ISBN|0-471-02403-1}}.
* {{cite journal | last = Shevtsova | first = I. G.
  | title = On the absolute constant in the Berry–Esseen inequality
  | year = 2008
  | journal = The Collection of Papers of Young Scientists of the Faculty of Computational Mathematics and Cybernetics 
  | issue = 5
  | pages = 101–110
  }}
* {{cite journal | last = Shevtsova
  | author-link = Irina Shevtsova
  | first = Irina
  | title = Sharpening of the upper bound of the absolute constant in the Berry–Esseen inequality
  | year = 2007
  | journal = Theory of Probability and Its Applications 
  | volume = 51 | issue = 3 
  | pages = 549–553
  | doi= 10.1137/S0040585X97982591
  }}
* {{cite journal | last = Shevtsova | first = Irina
  | title = An Improvement of Convergence Rate Estimates in the Lyapunov Theorem
  | year = 2010
  | journal = Doklady Mathematics 
  | volume = 82 | issue = 3 
  | pages = 862–864
  | doi= 10.1134/S1064562410060062
  | s2cid = 122973032 }}
* {{cite arXiv | last = Shevtsova | first = Irina
  | title = On the absolute constants in the Berry Esseen type inequalities for identically distributed summands
  | year = 2011
  | eprint = 1111.6554
  | class = math.PR
  }}
* {{cite journal | last = Shiganov | first = I.S.
  | title = Refinement of the upper bound of a constant in the remainder term of the central limit theorem
  | year = 1986
  | journal = Journal of Soviet Mathematics 
  | volume = 35
  | pages = 109–115
  | doi=10.1007/BF01121471 | issue=3
  | s2cid = 120112396 | doi-access = free
  }}
* {{cite journal | last = Tyurin | first = I.S.
  | title = On the accuracy of the Gaussian approximation
  | year = 2009
  | journal = Doklady Mathematics 
  | volume = 80 | issue = 3
  | pages = 840–843 | doi=10.1134/S1064562409060155
  | s2cid = 121383741 }}
* {{cite journal | last = Tyurin | first = I.S.
  | title = An improvement of upper estimates of the constants in the Lyapunov theorem
  | year = 2010
  | journal = Russian Mathematical Surveys 
  | volume = 65 | issue = 3(393)
  | pages = 201–202
  | doi = 10.1070/RM2010v065n03ABEH004688 | s2cid = 118771013
 }}
* {{cite journal | last = van Beek | first = P.
  | title = An application of Fourier methods to the problem of sharpening the Berry–Esseen inequality
  | year = 1972
  | journal = Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 
  | volume = 23
  | pages = 187–196
  | doi=10.1007/BF00536558 | issue=3
  | s2cid = 121036017 | doi-access = free
  }}
* {{cite journal | last = Zolotarev | first = V. M.
  | title = A sharpening of the inequality of Berry–Esseen
  | year = 1967
  | journal = Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
  | volume = 8
  | pages = 332–342 | doi=10.1007/BF00531598 | issue=4
  | s2cid = 122347713 | doi-access = free
  }}
{{refend}}

==External links==
* Gut, Allan & Holst Lars. [https://web.archive.org/web/20040202103426/http://www.stat.unipd.it/bernoulli/02a/bn_3.html Carl-Gustav Esseen], retrieved Mar. 15, 2004.
* {{springer|title=Berry–Esseen inequality|id=p/b015760}}

{{DEFAULTSORT:Berry-Esseen theorem}}
[[Category:Probabilistic inequalities]]
[[Category:Theorems in statistics]]
[[Category:Central limit theorem]]