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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with $E[|X_1|] \u003c \\infty$ and $\\mu = E[X_1]$. Let $\\bar{X}_n = (X_1 + \\dots + X_n)/n$.\nThen $\\bar{X}_n \\to \\mu$ [[in probability|0EFDDXTVD6]]: for each $\\varepsilon \u003e 0$,\n$$\\lim_{n \\to \\infty} P\\left(\\left|\\bar{X}_n - \\mu\\right| \u003e \\varepsilon\\right) = 0.$$\n\n## Remarks\n\nIf $\\sigma^2 = \\operatorname{Var}(X_1) \u003c \\infty$, there is a short proof. By the [[variance of a sum|0V0778DCW9]], $\\operatorname{Var}(\\bar{X}_n) = \\sigma^2/n$.\nThen [[Chebyshev's inequality|02HZ1BK3MW]] gives $P(|\\bar{X}_n - \\mu| \u003e \\varepsilon) \\le \\sigma^2 / (n \\varepsilon^2) \\to 0$.\nThe general case uses truncation, or characteristic functions: $\\varphi_{\\bar{X}_n}(t) = \\varphi_{X_1}(t/n)^n \\to e^{i\\mu t}$.\nThe finite-variance proof needs only that the $X_i$ are pairwise uncorrelated with the same mean and bounded variances.\n\n## History\n\nJacob Bernoulli proved the first version, for Bernoulli trials. It appeared after his death in Ars Conjectandi in 1713.\n\n## Explanations\n\n### L1 human\n\nThe weak law says that for a large number $n$ of trials, the sample mean is very likely to be close to the true mean.\nFor example, in $10000$ tosses of a fair coin, the fraction of heads is very likely to be between $0.49$ and $0.51$.\n",
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  "name": "Weak law of large numbers",
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