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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[i.i.d.|00MRN0FARV]] real random variables with $E[X_1] = 0$, $\\sigma^2 = E[X_1^2] \u003e 0$ and $\\rho = E[\\abs{X_1}^3] \u003c \\infty$.\nLet $F_n$ 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]].\nThen there is an absolute constant $C$, independent of $n$ and of the law of $X_1$, such that for all $n \\ge 1$,\n$$\\sup_{x \\in \\R} \\abs{F_n(x) - \\Phi(x)} \\le \\frac{C \\rho}{\\sigma^3 \\sqrt{n}}.$$\n\n## Remarks\n\nThe theorem gives a rate for the [[central limit theorem|0EPZC4Z1FC]], which by itself only says that the left side tends to $0$.\nThe 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.\nThe best value of $C$ is not known; it is known that one may take $C \u003c 1/2$.\n\nThe rate $n^{-1/2}$ cannot be improved in general.\nLet $X_k$ be $1$ or $-1$ with probability $1/2$ each, so $\\sigma = \\rho = 1$.\nFor 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 $n^{-1/2}$.\nSo $F_n$ has a jump of that size at $0$, while $\\Phi$ is continuous, and the supremum is at least half the jump.\n\n## Explanations\n\n### L1 human\n\nThe central limit theorem says that a standardised sum of many independent copies of a variable has nearly a normal distribution.\nThe Berry–Esseen theorem says how near: the error is at most a constant times $1/\\sqrt{n}$.\nThe bound depends on the law of the variable only through the ratio of its third absolute moment to the cube of its standard deviation.\n\n### L2 human\n\nThe bound turns the normal approximation into a quantitative statement, which matters in statistics when the sample size is fixed.\nIt needs a finite third [[moment|01XAHPADBA]]; with only a finite variance, the convergence in the central limit theorem can be arbitrarily slow.\nThere 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.\nStein's method gives another proof and extends to dependent variables.\n",
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