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  "body": "\n## Summary\n\nA stable distribution is a law on $\\R$ whose type is preserved under independent sums.\nA sum of independent copies has the same law up to a change of scale and location.\nEach stable law has an index $\\alpha \\in (0, 2]$; the [[normal distribution|05AQE0QR0D]] is the case $\\alpha = 2$ and the [[Cauchy distribution|0GPG24Q3GR]] is a case with $\\alpha = 1$.\nStable laws are exactly the possible nondegenerate limits of suitably normalised sums of i.i.d. random variables, which extends the [[central limit theorem|0EPZC4Z1FC]] to variables of infinite variance.\nEvery stable law is [[infinitely divisible|BX3V3NYM1Y]].\n\n## Explanations\n\n### L1 human\n\nAdd two independent normal variables and you get a normal variable again, only wider; stable laws are the laws with this property.\nFor an index $\\alpha \u003c 2$ the tails are heavy: the chance that $\\abs{X}$ exceeds $x$ falls like $x^{-\\alpha}$, and the variance is infinite.\nFor example, the average of $n$ independent standard Cauchy variables is again standard Cauchy, so averaging does not reduce the spread at all.\nStable laws appear as limits whenever many independent heavy-tailed effects add up.\n",
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