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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with mean $\\mu$ and variance $\\sigma^2$, where $0 \u003c \\sigma^2 \u003c \\infty$. Let $\\bar{X}_n = (X_1 + \\dots + X_n)/n$.\nThen $\\sqrt{n} \\, (\\bar{X}_n - \\mu) / \\sigma$ converges [[in distribution|0JDAQHWA9N]] to $\\mathcal{N}(0, 1)$. That is, for each $x \\in \\mathbb{R}$,\n$$\\lim_{n \\to \\infty} P\\left(\\frac{X_1 + \\dots + X_n - n\\mu}{\\sigma \\sqrt{n}} \\le x\\right) = \\Phi(x) = \\int_{-\\infty}^{x} \\frac{1}{\\sqrt{2\\pi}} e^{-u^2/2} \\, du.$$\n\n## Remarks\n\nA standard proof uses [[characteristic functions|0MYDNRT7PP]]. With $Y_i = (X_i - \\mu)/\\sigma$, $\\varphi_{Y_1}(t) = 1 - t^2/2 + o(t^2)$, so $\\varphi_{Y_1}(t/\\sqrt{n})^n \\to e^{-t^2/2}$.\n[[Lévy's continuity theorem|0FPWF7TTMB]] then gives the convergence in distribution.\nThe Lindeberg and Lyapunov forms remove the condition that the $X_i$ have the same law. The Berry–Esseen theorem bounds the rate of convergence when $E[|X_1|^3] \u003c \\infty$.\nThe finite variance is necessary. Sums of i.i.d. Cauchy variables, scaled by $1/n$, stay Cauchy.\n\n## Explanations\n\n### L1 human\n\nThe central limit theorem explains why the bell curve is everywhere.\nA sum of many small independent effects is approximately normal, whatever the law of each effect, as long as the variance is finite.\n",
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