---
tag: H9KTH7B380
kind: statement
role: theorem
name: Berry–Esseen theorem
msc: [60F05, "60E15"]
uses: [00MRN0FARV, 0H8H7ZTHW2, 05AQE0QR0D, 01XAHPADBA, 0477R9PYAE, 0EPZC4Z1FC]
about: [0EPZC4Z1FC, 0H8H7ZTHW2]
sources:
  - url: https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem
same_as:
  wikidata: Q1785872
  wikipedia: Berry–Esseen theorem
---

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables with $E[X_1] = 0$, $\sigma^2 = E[X_1^2] > 0$ and $\rho = E[\abs{X_1}^3] < \infty$.
Let $F_n$ be the [[cumulative distribution function|0H8H7ZTHW2]] of $(X_1 + \cdots + X_n) / (\sigma \sqrt{n})$, and let $\Phi$ be the cumulative distribution function of the standard [[normal distribution|05AQE0QR0D]].
Then there is an absolute constant $C$, independent of $n$ and of the law of $X_1$, such that for all $n \ge 1$,
$$\sup_{x \in \R} \abs{F_n(x) - \Phi(x)} \le \frac{C \rho}{\sigma^3 \sqrt{n}}.$$

## Remarks

The theorem gives a rate for the [[central limit theorem|0EPZC4Z1FC]], which by itself only says that the left side tends to $0$.
The classical proof compares characteristic functions and uses Esseen's smoothing inequality, which bounds the distance between two distribution functions by an integral of the difference of their characteristic functions.
The best value of $C$ is not known; it is known that one may take $C < 1/2$.

The rate $n^{-1/2}$ cannot be improved in general.
Let $X_k$ be $1$ or $-1$ with probability $1/2$ each, so $\sigma = \rho = 1$.
For even $n$, the variable $S_n = X_1 + \cdots + X_n$ takes the value $0$ with probability $\binom{n}{n/2} 2^{-n}$, which is of order $n^{-1/2}$.
So $F_n$ has a jump of that size at $0$, while $\Phi$ is continuous, and the supremum is at least half the jump.

## Explanations

### L1 human

The central limit theorem says that a standardised sum of many independent copies of a variable has nearly a normal distribution.
The Berry–Esseen theorem says how near: the error is at most a constant times $1/\sqrt{n}$.
The bound depends on the law of the variable only through the ratio of its third absolute moment to the cube of its standard deviation.

### L2 human

The bound turns the normal approximation into a quantitative statement, which matters in statistics when the sample size is fixed.
It needs a finite third [[moment|01XAHPADBA]]; with only a finite variance, the convergence in the central limit theorem can be arbitrarily slow.
There are versions for independent variables that are not identically distributed, with $\sum_k E[\abs{X_k}^3] / s_n^3$ in place of $\rho / (\sigma^3 \sqrt{n})$, and multivariate versions.
Stein's method gives another proof and extends to dependent variables.