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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[independent and identically distributed|0AJN7WHBJZ]] real random variables with mean $\\mu$, variance $\\sigma^2 \u003e 0$ and $E\\abs{X_1}^3 \u003c \\infty$.\nLet $\\mu_3 = E\\big((X_1 - \\mu)^3\\big)$ be the third [[central moment|0ZT24EC90K]].\nAssume that the distribution of $X_1$ is not a lattice distribution.\nThat is, it is not concentrated on a set of the form $\\set{a + kh : k \\in \\Z}$ with $h \u003e 0$.\nLet $F_n$ be the distribution function of $S_n = \\big(\\sum_{i=1}^n X_i - n\\mu\\big)/(\\sigma\\sqrt{n})$, and let $\\Phi$ and $\\varphi$ be the distribution function and the density of the standard [[normal distribution|07RXBVTVZB]].\nThen\n$$F_n(x) = \\Phi(x) - \\frac{\\mu_3}{6 \\sigma^3 \\sqrt{n}} (x^2 - 1)\\, \\varphi(x) + o\\!\\left(\\frac{1}{\\sqrt{n}}\\right)$$\nuniformly in $x \\in \\R$.\n\n## Remarks\n\nThe proof compares characteristic functions.\nThe characteristic function of $S_n$ is $e^{-t^2/2}\\big(1 + \\frac{\\mu_3}{6\\sigma^3\\sqrt{n}}(it)^3\\big) + o(n^{-1/2})$ for $t$ in ranges of order $\\sqrt{n}$, and the correction term is the Fourier transform of the correction in the statement.\nA smoothing inequality turns closeness of characteristic functions into closeness of distribution functions; the non-lattice hypothesis controls the characteristic function away from $0$.\n\nThe [[central limit theorem|0EPZC4Z1FC]] says $F_n(x) \\to \\Phi(x)$; the expansion gives the first correction, of order $1/\\sqrt{n}$, which comes from skewness.\nFor a symmetric distribution, $\\mu_3 = 0$ and the error of the normal approximation is $o(1/\\sqrt{n})$.\nUnder stronger moment conditions and Cramér's condition $\\limsup_{\\abs{t} \\to \\infty} \\abs{E e^{itX_1}} \u003c 1$, one gets further terms in powers of $1/\\sqrt{n}$ with Hermite polynomials and higher cumulants.\n\nThe non-lattice hypothesis cannot be dropped.\nFor Bernoulli($1/2$) variables, $F_n$ has jumps of size of order $1/\\sqrt{n}$, while the right side is continuous in $x$, so the error cannot be $o(1/\\sqrt{n})$ uniformly.\n\n## Explanations\n\n### L1 human\n\nThe central limit theorem says that a standardized sum is approximately normal.\nThe Edgeworth expansion says how the first error looks.\nIf the summands are skewed, the distribution of the sum is a little skewed too, by an amount that shrinks like $1/\\sqrt{n}$.\nThe formula gives the shape of this correction.\n\n### L2 human\n\nThe Berry–Esseen theorem bounds the error of the normal approximation by $C\\, E\\abs{X_1 - \\mu}^3/(\\sigma^3\\sqrt{n})$; the Edgeworth expansion identifies the leading term of that error.\nExpansions of this kind explain why the bootstrap is often more accurate than the normal approximation for studentized statistics.\nThey also give corrected quantiles, through the Cornish–Fisher expansion, and corrected confidence intervals.\n",
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