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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 $\\bar{X}_n \\to \\mu$ [[almost surely|04P7D63W8P]]:\n$$P\\left(\\lim_{n \\to \\infty} \\bar{X}_n = \\mu\\right) = 1.$$\n\n## Remarks\n\nThe strong law implies the [[weak law|0BFDHF2H7N]], because almost sure convergence implies convergence in probability.\nA short proof exists if $E[X_1^4] \u003c \\infty$: then $E[(\\bar{X}_n - \\mu)^4] \\le C/n^2$, and the first Borel–Cantelli lemma applies.\nEtemadi proved that the result holds when the $X_i$ are only pairwise independent and identically distributed.\nConversely, if $E[|X_1|] = \\infty$, then almost surely $\\bar{X}_n$ does not converge to a finite limit.\n\n## Explanations\n\n### L1 human\n\nThe strong law says more than the weak law. Almost every infinite sequence of trials has sample means that converge to the true mean.\nSo, with probability $1$, the fraction of heads in an infinite sequence of fair coin tosses tends to $1/2$.\n",
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