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 n^−1/2, where n is the sample size, and the constant is estimated in terms of the third absolute normalized moment. It is also possible to give non-uniform bounds which become more strict for more extreme events.

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.

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

There exists a positive constant C such that if X₁, X₂, ..., are i.i.d. random variables with E(X₁) = 0, E(X₁²) = σ² > 0, and E(|X₁|³) = ρ < ∞,^{[note 1]} and if we define

${\displaystyle Y_{n}={X_{1}+X_{2}+\cdots +X_{n} \over n}}$

the sample mean, with F_n the cumulative distribution function of

${\displaystyle {Y_{n}{\sqrt {n}} \over {\sigma }},}$

and Φ the cumulative distribution function of the standard normal distribution, then for all x and n,

${\displaystyle \left|F_{n}(x)-\Phi (x)\right|\leq {C\rho \over \sigma ^{3}{\sqrt {n}}}.\ \ \ \ (1)}$

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 upper bounds on the constant C have decreased markedly over the years, from the original value of 7.59 by Esseen in 1942.^[1] The estimate C < 0.4748 follows from the inequality

${\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq {0.33554(\rho +0.415\sigma ^{3}) \over \sigma ^{3}{\sqrt {n}}},}$

since σ³ ≤ ρ and 0.33554 · 1.415 < 0.4748. However, if ρ ≥ 1.286σ³, then the estimate

${\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq {0.3328(\rho +0.429\sigma ^{3}) \over \sigma ^{3}{\sqrt {n}}},}$

is even tighter.^[2]

Esseen (1956) proved that the constant also satisfies the lower bound

${\displaystyle C\geq {\frac {{\sqrt {10}}+3}{6{\sqrt {2\pi }}}}\approx 0.40973\approx {\frac {1}{\sqrt {2\pi }}}+0.01079.}$

Let X₁, X₂, ..., be independent random variables with E(X_i) = 0, E(X_i²) = σ_i² > 0, and E(|X_i|³) = ρ_i < ∞. Also, let

${\displaystyle S_{n}={X_{1}+X_{2}+\cdots +X_{n} \over {\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}+\cdots +\sigma _{n}^{2}}}}}$

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

${\displaystyle {\vec {\sigma }}=(\sigma _{1},\ldots ,\sigma _{n}),\ {\vec {\rho }}=(\rho _{1},\ldots ,\rho _{n}).}$

In 1941, Andrew C. Berry proved that for all n there exists an absolute constant C₁ such that

${\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq C_{1}\cdot \psi _{1},\ \ \ \ (2)}$

where

${\displaystyle \psi _{1}=\psi _{1}{\big (}{\vec {\sigma }},{\vec {\rho }}{\big )}={\Big (}{\textstyle \sum \limits _{i=1}^{n}\sigma _{i}^{2}}{\Big )}^{-1/2}\cdot \max _{1\leq i\leq n}{\frac {\rho _{i}}{\sigma _{i}^{2}}}.}$

Independently, in 1942, Carl-Gustav Esseen proved that for all n there exists an absolute constant C₀ such that

${\displaystyle \sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq C_{0}\cdot \psi _{0},\ \ \ \ (3)}$

where

${\displaystyle \psi _{0}=\psi _{0}{\big (}{\vec {\sigma }},{\vec {\rho }}{\big )}={\Big (}{\textstyle \sum \limits _{i=1}^{n}\sigma _{i}^{2}}{\Big )}^{-3/2}\cdot \sum \limits _{i=1}^{n}\rho _{i}.}$

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 X₁, ..., X_n have identical distributions

${\displaystyle \psi _{0}=\psi _{1}={\frac {\rho _{1}}{\sigma _{1}^{3}{\sqrt {n}}}},}$

and thus the bounds stated by inequalities (1), (2) and (3) coincide apart from the constant.

Regarding C₀, obviously, the lower bound established by Esseen (1956) remains valid:

${\displaystyle C_{0}\geq {\frac {{\sqrt {10}}+3}{6{\sqrt {2\pi }}}}=0.4097\ldots .}$

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

The upper bounds for C₀ were subsequently lowered from Esseen's original estimate 7.59 to 0.5600.^[3]

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.^[4] It is only a portion of the results that they established:

Let ${\displaystyle \{X_{i}\}}$ be independent, identically distributed random variables with ${\displaystyle E(X_{i})=\mu }$, ${\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}}$, ${\displaystyle E|X_{i}-\mu |^{3}=\kappa ^{3}}$. Let ${\displaystyle N}$ be a non-negative integer-valued random variable, independent from ${\displaystyle \{X_{i}\}}$. Let ${\displaystyle S_{N}=X_{1}+\cdots +X_{N}}$, and define

${\displaystyle \Delta =\sup _{x}\left|P\left({\frac {S_{N}-E(S_{N})}{\sqrt {\operatorname {Var} (S_{N})}}}\leq z\right)-\Phi (z)\right|}$

Then

${\displaystyle \Delta \leq 3.8696{\frac {\kappa ^{3}}{{\sqrt {E(N)}}\sigma ^{3}}}+1.0395{\frac {E|N-E(N)|}{E(N)}}+0.2420{\frac {\mu ^{2}\operatorname {Var} (N)}{\sigma ^{2}E(N)}}}$

As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.^[5][6]

Let ${\displaystyle X_{1},\dots ,X_{n}}$ be independent ${\displaystyle \mathbb {R} ^{d}}$-valued random vectors each having mean zero. Write ${\displaystyle S_{n}=\sum _{i=1}^{n}X_{i}}$ and assume ${\displaystyle \Sigma _{n}=\operatorname {Cov} [S_{n}]}$ is invertible. Let ${\displaystyle Z_{n}\sim \operatorname {N} (0,{\Sigma _{n}})}$ be a ${\displaystyle d}$-dimensional Gaussian with the same mean and covariance matrix as ${\displaystyle S_{n}}$. Then for all convex sets ${\displaystyle U\subseteq \mathbb {R} ^{d}}$,

${\displaystyle {\big |}\Pr[S_{n}\in U]-\Pr[Z_{n}\in U]\,{\big |}\leq Cd^{1/4}\gamma _{n}}$,

where ${\displaystyle C}$ is a universal constant and ${\displaystyle \gamma _{n}=\sum _{i=1}^{n}\operatorname {E} {\big [}\|\Sigma _{n}^{-1/2}X_{i}\|_{2}^{3}{\big ]}}$ (the third power of the L² norm).

The dependency on ${\displaystyle d^{1/4}}$ is conjectured to be optimal, but might not be.^[6]

The bounds given above consider the maximal difference between the cdf's. They are 'uniform' in that they do not depend on ${\displaystyle x}$ and quantify the uniform convergence ${\displaystyle F_{n}\to \Phi }$. However, because ${\displaystyle F_{n}(x)-\Phi (x)}$ goes to zero for large ${\displaystyle x}$ 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 ${\displaystyle x}$ and in this way become smaller for large ${\displaystyle x}$.

One such result going back to (Esseen 1945) that was since improved multiple times is the following.

As above, let X₁, X₂, ..., be independent random variables with E(X_i) = 0, E(X_i²) = σ_i² > 0, and E(|X_i|³) = ρ_i < ∞. Also, let ${\displaystyle \sigma ^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}}$ and

${\displaystyle S_{n}={X_{1}+X_{2}+\cdots +X_{n} \over \sigma }}$

be the normalized n-th partial sum. Denote F_n the cdf of S_n, and Φ the cdf of the standard normal distribution. Then

${\displaystyle |F_{n}(x)-\Phi (x)|\leq {\frac {C_{3}}{\sigma ^{3}+|x|^{3}}}\cdot \sum _{i=1}^{n}\rho _{i}}$,

where ${\displaystyle C_{3}}$ is a universal constant.

The constant ${\displaystyle C_{3}}$ may be taken as 114.667.^[7] Moreover, if the ${\displaystyle X_{i}}$ are identically distributed, it can be taken as ${\displaystyle C+8(1+\mathrm {e} )}$, where ${\displaystyle C}$ is the constant from the first theorem above, and hence 30.2211 works.^[8]

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

• Gut, Allan & Holst Lars. Carl-Gustav Esseen, retrieved Mar. 15, 2004.
• "Berry–Esseen inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]