# 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.

## 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$_1$$^2$) = σ$^2$ > 0, and E(|X$_1$|$^3$) = ρ < ∞, and if we define
       $Y_{n}={X_{1}+X_{2}+\cdots +X_{n} \over n}$
    the sample mean, with F$_n$ the cumulative distribution function of
       ${Y_{n}{\sqrt {n}} \over {\sigma }},$
    and Φ the cumulative distribution function of the standard normal distribution, then for all x and n,
       $\left|F_{n}(x)-\Phi (x)\right|\leq {C\rho \over \sigma ^{3}{\sqrt {n}}}.\ \ \ \ (1)$
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. The estimate C < 0.4748 follows from the inequality
    $\sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq {0.33554(\rho +0.415\sigma ^{3}) \over \sigma ^{3}{\sqrt {n}}},$
since σ$^3$ ≤ ρ and 0.33554 · 1.415 < 0.4748. However, if ρ ≥ 1.286σ$^3$, then the estimate
    $\sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq {0.3328(\rho +0.429\sigma ^{3}) \over \sigma ^{3}{\sqrt {n}}},$
is even tighter.
Esseen (1956) proved that the constant also satisfies the lower bound
    $C\geq {\frac {{\sqrt {10}}+3}{6{\sqrt {2\pi }}}}\approx 0.40973\approx {\frac {1}{\sqrt {2\pi }}}+0.01079.$

### 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
       $S_{n}={X_{1}+X_{2}+\cdots +X_{n} \over {\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}+\cdots +\sigma _{n}^{2}}}}$
    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
       ${\vec {\sigma }}=(\sigma _{1},\ldots ,\sigma _{n}),\ {\vec {\rho }}=(\rho _{1},\ldots ,\rho _{n}).$
    In 1941, Andrew C. Berry proved that for all n there exists an absolute constant C$_1$ such that
       $\sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq C_{1}\cdot \psi _{1},\ \ \ \ (2)$
    where
       $\psi _{1}=\psi _{1}{\big{\vec {\sigma }},{\vec {\rho }}{\big )}={\Big{\textstyle \sum \limits _{i=1}^{n}\sigma _{i}^{2}}{\Big )}^{-1/2}\cdot \max _{1\leq i\leq n}{\frac {\rho _{i}}{\sigma _{i}^{2}}}.$
    Independently, in 1942, Carl-Gustav Esseen proved that for all n there exists an absolute constant C$_0$ such that
       $\sup _{x\in \mathbb {R} }\left|F_{n}(x)-\Phi (x)\right|\leq C_{0}\cdot \psi _{0},\ \ \ \ (3)$
    where
       $\psi _{0}=\psi _{0}{\big{\vec {\sigma }},{\vec {\rho }}{\big )}={\Big{\textstyle \sum \limits _{i=1}^{n}\sigma _{i}^{2}}{\Big )}^{-3/2}\cdot \sum \limits _{i=1}^{n}\rho _{i}.$
It is easy to make sure that ψ$_0$≤ψ$_1$. Due to this circumstance inequality (3) is conventionally called the Berry–Esseen inequality, and the quantity ψ$_0$ is called the Lyapunov fraction of the third order. Moreover, in the case where the summands X$_1$, ..., X$_n$ have identical distributions
       $\psi _{0}=\psi _{1}={\frac {\rho _{1}}{\sigma _{1}^{3}{\sqrt {n}}}},$
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 Esseen (1956) remains valid:
    $C_{0}\geq {\frac {{\sqrt {10}}+3}{6{\sqrt {2\pi }}}}=0.4097\ldots .$
The lower bound is exactly reached only for certain Bernoulli distributions (see Esseen (1956) for their explicit expressions).
The upper bounds for C$_0$ 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 $\{X_{i}\}$ be independent, identically distributed random variables with $E(X_{i})=\mu$, $\operatorname {Var} (X_{i})=\sigma ^{2}$, $E|X_{i}-\mu |^{3}=\kappa ^{3}$. Let $N$ be a non-negative integer-valued random variable, independent from $\{X_{i}\}$. Let $S_{N}=X_{1}+\cdots +X_{N}$, and define
       $\Delta =\sup _{x}\left|P\left({\frac {S_{N}-E(S_{N})}{\sqrt {\operatorname {Var} (S_{N})}}}\leq z\right)-\Phi (z)\right|$
    Then
       $\Delta \leq 3.8696{\frac {\kappa ^{3}}{{\sqrt {E(N)}}\sigma ^{3}}}+1.0395{\frac {E|N-E(N)|}{E(N)}}+0.2420{\frac {\mu ^{2}\operatorname {Var} (N)}{\sigma ^{2}E(N)}}$

### Multidimensional version

As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.
    Let $X_{1},\dots ,X_{n}$ be independent $\mathbb {R} ^{d}$-valued random vectors each having mean zero. Write $S_{n}=\sum _{i=1}^{n}X_{i}$ and assume $\Sigma _{n}=\operatorname {Cov} [S_{n}]$ is invertible. Let $Z_{n}\sim \operatorname {N} (0,{\Sigma _{n}})$ be a $d$-dimensional Gaussian with the same mean and covariance matrix as $S_{n}$. Then for all convex sets $U\subseteq \mathbb {R} ^{d}$,
       ${\big |}\Pr[S_{n}\in U]-\Pr[Z_{n}\in U]\,{\big |}\leq Cd^{1/4}\gamma _{n}$,
    where $C$ is a universal constant and $\gamma _{n}=\sum _{i=1}^{n}\operatorname {E} {\big\|\Sigma _{n}^{-1/2}X_{i}\|_{2}^{3}{\big ]}$ (the third power of the L$^2$ norm).
The dependency on $d^{1/4}$ is conjectured to be optimal, but might not 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 $x$ and quantify the uniform convergence $F_{n}\to \Phi$. However, because $F_{n}(x)-\Phi (x)$ goes to zero for large $x$ 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 $x$ and in this way become smaller for large $x$.
One such result going back to (Esseen 1945) that was since improved multiple times is the following.
    As above, 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 $\sigma ^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}$ and
       $S_{n}={X_{1}+X_{2}+\cdots +X_{n} \over \sigma }$
    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
       $|F_{n}(x)-\Phi (x)|\leq {\frac {C_{3}}{1+|x|^{3}}}\cdot \sigma ^{-3}\sum _{i=1}^{n}\rho _{i}$,
    where $C_{3}$ is a universal constant.
The constant $C_{3}$ may be taken as 114.667. Moreover, if the $X_{i}$ are identically distributed, it can be taken as $C+8(1+\mathrm {e} )$, where $C$ is the constant from the first theorem above, and hence 30.2211 works.