Berry–Esseen theorem

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 X₁, X₂, ..., are i.i.d. random variables with E(X₁) = 0, E(X₁²) = σ² > 0, and E(|X₁|³) = ρ < ∞, and if we define

${\displaystyle Y_n=\frac{X_1+X_2+\cdots +X_n}{n}}$

the sample mean, with F_n the cumulative distribution function of

${\displaystyle \frac{Y_n\sqrt{n}}{\sigma },}$

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

${\displaystyle \left|F_n(x)-\mathrm{\Phi }(x)\right|\le \frac{C\rho }{{\sigma }^3\sqrt{n}}.(1)}$

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. The estimate C < 0.4748 follows from the inequality

${\displaystyle \underset{x\in \mathbb{R}}{sup}\left|F_n(x)-\mathrm{\Phi }(x)\right|\le \frac{0.33554(\rho +0.415{\sigma }^3)}{{\sigma }^3\sqrt{n}},}$

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

${\displaystyle \underset{x\in \mathbb{R}}{sup}\left|F_n(x)-\mathrm{\Phi }(x)\right|\le \frac{0.3328(\rho +0.429{\sigma }^3)}{{\sigma }^3\sqrt{n}},}$

is even tighter.

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

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

Non-identically distributed summands

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

${\displaystyle S_n=\frac{X_1+X_2+\cdots +X_n}{\sqrt{{\sigma }_1^2+{\sigma }_2^2+\cdots +{\sigma }_n^2}}}$

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

${\displaystyle \overrightarrow{\sigma }=({\sigma }_1,\dots ,{\sigma }_n),\overrightarrow{\rho }=({\rho }_1,\dots ,{\rho }_n).}$

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

${\displaystyle \underset{x\in \mathbb{R}}{sup}\left|F_n(x)-\mathrm{\Phi }(x)\right|\le C_1\cdot {\psi }_1,(2)}$

where

${\displaystyle {\psi }_1={\psi }_1(\overrightarrow{\sigma },\overrightarrow{\rho })=(\sum _{i=1}^n{\sigma }_i^2{)}^{-1/2}\cdot \underset{1\le i\le n}{max}\frac{{\rho }_i}{{\sigma }_i^2}.}$

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

${\displaystyle \underset{x\in \mathbb{R}}{sup}\left|F_n(x)-\mathrm{\Phi }(x)\right|\le C_0\cdot {\psi }_0,(3)}$

where

${\displaystyle {\psi }_0={\psi }_0(\overrightarrow{\sigma },\overrightarrow{\rho })=(\sum _{i=1}^n{\sigma }_i^2{)}^{-3/2}\cdot \sum _{i=1}^n{\rho }_i.}$

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:

${\displaystyle C_0\ge \frac{\sqrt{10}+3}{6\sqrt{2\pi }}=0.4097\dots .}$

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.

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. It is only a portion of the results that they established:

Let ${\displaystyle \{X_i\}}$ be independent, identically distributed random variables with ${\displaystyle E(X_i)=\mu }$, ${\displaystyle \mathrm{Var}(X_i)={\sigma }^2}$, ${\displaystyle E|X_i-\mu {|}^3={\kappa }^3}$. 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

${\displaystyle \mathrm{\Delta }=\underset{x}{sup}\left|P\left(\frac{S_N-E(S_N)}{\sqrt{\mathrm{Var}(S_N)}}\le z\right)-\mathrm{\Phi }(z)\right|.}$

Then

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

Multidimensional version

As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.

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 ${\displaystyle {\mathrm{\Sigma }}_n=\mathrm{Cov}[S_n]}$ is invertible. Let ${\displaystyle Z_n\sim \mathrm{N}(0,{\mathrm{\Sigma }}_n)}$ be a ${\displaystyle d}$-dimensional Gaussian with the same mean and covariance matrix as ${\displaystyle S_n}$. Then for all convex sets ${\displaystyle U\subseteq {\mathbb{R}}^d}$,

${\displaystyle |Pr[S_n\in U]-Pr[Z_n\in U]|\le C{d}^{1/4}{\gamma }_n,}$

where ${\displaystyle C}$ is a universal constant and ${\displaystyle {\gamma }_n=\sum _{i=1}^n\mathrm{E}[\Vert {\mathrm{\Sigma }}_n^{-1/2}{X}_i{\Vert }_2^3]}$ (the third power of the L² norm).

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