Berry–Esseen theorem

In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample 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 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 independent samples, the convergence rate is , where is the sample size, and the constant is estimated in terms of the third absolute normalized moment.

Statement of the theorem

Statements of the theorem vary, as it was independently discovered by two mathematicians, 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 ''C'' such that if ''X''₁, ''X''₂, ..., are i.i.d. random variables with E(''X''₁) = 0, E(''X''₁²) = σ² > 0, and E(, ''X''₁, ³) = ρ < ∞,Since the random variables are identically distributed, ''X''₂, ''X''₃, ... all have the same moments as ''X''₁. and if we define
::$Y_n =$
:the sample mean, with ''F''_''n'' the cumulative distribution function of
::$,$
:and Φ the cumulative distribution function of the standard normal distribution, then for all ''x'' and ''n'',
::$\left, F_n(x) - \Phi(x)\ \le .\ \ \ \ (1)$

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 is finite, then the cumulative distribution functions of the 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 order of ''n''^−1/2.

Calculated values of the constant ''C'' have decreased markedly over the years, from the original value of 7.59 by , to 0.7882 by , then 0.7655 by , then 0.7056 by , then 0.7005 by , then 0.5894 by , then 0.5129 by , then 0.4785 by . The detailed review can be found in the papers and . The best estimate , ''C'' < 0.4748, follows from the inequality
:$\sup_\left, F_n(x) - \Phi(x)\ \le ,$
due to , since σ³ ≤ ρ and 0.33554 · 1.415 < 0.4748. However, if ρ ≥ 1.286σ³, then the estimate
:$\sup_\left, F_n(x) - \Phi(x)\ \le ,$
which is also proved in , gives an even tighter upper estimate.

proved that the constant also satisfies the lower bound
: $C\geq\frac \approx 0.40973 \approx \frac + 0.01079 .$

Non-identically distributed summands

:Let ''X''₁, ''X''₂, ..., be independent random variables with E(''X''_''i'') = 0, E(''X''_''i''²) = σ_''i''² > 0, and E(, ''X''_''i'', ³) = ρ_''i'' < ∞. Also, let
::$S_n =$
:be the normalized ''n''-th partial sum. Denote ''F''_''n'' the cdf of ''S''_''n'', and Φ the cdf of the standard normal distribution. For the sake of convenience denote
::$\vec=(\sigma_1,\ldots,\sigma_n),\ \vec=(\rho_1,\ldots,\rho_n).$
:In 1941, Andrew C. Berry proved that for all ''n'' there exists an absolute constant ''C''₁ such that
::$\sup_\left, F_n(x) - \Phi(x)\ \le C_1\cdot\psi_1,\ \ \ \ (2)$
:where
::$\psi_1=\psi_1\big(\vec,\vec\big)=\Big(\Big)^\cdot\max_\frac.$

:Independently, in 1942, Carl-Gustav Esseen proved that for all ''n'' there exists an absolute constant ''C''₀ such that
::$\sup_\left, F_n(x) - \Phi(x)\ \le C_0\cdot\psi_0, \ \ \ \ (3)$
:where
::$\psi_0=\psi_0\big(\vec,\vec\big)=\Big(\Big)^\cdot\sum\limits_^n\rho_i.$

It is easy to make sure that ψ₀≤ψ₁. Due to this circumstance inequality (3) is conventionally called the Berry–Esseen inequality, and the quantity ψ₀ is called the Lyapunov fraction of the third order. Moreover, in the case where the summands ''X''₁, ..., ''X''_''n'' have identical distributions
::$\psi_0=\psi_1=\frac,$
and thus the bounds stated by inequalities (1), (2) and (3) coincide apart from the constant.

Regarding ''C''₀, obviously, the lower bound established by remains valid:
: $C_0\geq\frac = 0.4097\ldots.$

The upper bounds for ''C''₀ were subsequently lowered from the original estimate 7.59 due to to (considering recent results only) 0.9051 due to , 0.7975 due to , 0.7915 due to , 0.6379 and 0.5606 due to and . the best estimate is 0.5600 obtained by .

Multidimensional version

As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.Bentkus, Vidmantas. "A Lyapunov-type bound in R^d." Theory of Probability & Its Applications 49.2 (2005): 311–323.

Let $X_1,\dots,X_n$ be independent $\mathbb R^d$-valued random vectors each having mean zero. Write $S = \sum_^n X_i$ and assume \Sigma = \operatorname /math> is invertible. Let $Z\sim\operatorname(0,\Sigma)$ be a $d$-dimensional Gaussian with the same mean and covariance matrix as $S$. Then for all convex sets $U\subseteq\mathbb R^d$,
:\big, \Pr \in U\Pr \in U,\big, \le C d^ \gamma,
where $C$ is a universal constant and \gamma=\sum_^n \operatorname\big \Sigma^X_i\, _2^3\big/math> (the third power of the L² norm).

The dependency on $d^$ is conjectured to be optimal, but might not be.

See also

*Chernoff's inequality
*Edgeworth series
*List of inequalities
* List of mathematical theorems
* Concentration inequality

Notes

References

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* Durrett, Richard (1991). ''Probability: Theory and Examples''. Pacific Grove, CA: Wadsworth & Brooks/Cole. .
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* Feller, William (1972). ''An Introduction to Probability Theory and Its Applications, Volume II'' (2nd ed.). New York: John Wiley & Sons. .
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* Manoukian, Edward B. (1986). ''Modern Concepts and Theorems of Mathematical Statistics''. New York: Springer-Verlag. .
* Serfling, Robert J. (1980). ''Approximation Theorems of Mathematical Statistics''. New York: John Wiley & Sons. .
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External links

* Gut, Allan & Holst Lars
Carl-Gustav Esseen
retrieved Mar. 15, 2004.
*

{{DEFAULTSORT:Berry-Esseen theorem
Probabilistic inequalities
Theorems in statistics
Central limit theorem