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  "body": "\n## Definition\n\nLet $X$ be a real [[random variable|0PNRKMRC55]] that is not almost surely constant.\nLet $X_1, X_2$ be independent random variables with the same law as $X$.\nThe law of $X$ is *stable* if for all $a, b \u003e 0$ there are $c \u003e 0$ and $d \\in \\R$ such that $a X_1 + b X_2$ has the same law as $c X + d$.\nIt is *strictly stable* if moreover $d = 0$ can be chosen for all $a, b \u003e 0$.\n\n## Remarks\n\nFor a stable law there is a unique $\\alpha \\in (0, 2]$, the index, such that $c^\\alpha = a^\\alpha + b^\\alpha$ always.\nThe [[characteristic function|0V5B6D6BQH]] of a stable law has the form $\\varphi(t) = \\exp\\left(i \\mu t - \\abs{\\gamma t}^\\alpha (1 - i \\beta \\sgn(t) \\Phi(t))\\right)$, with $\\Phi(t) = \\tan(\\pi\\alpha/2)$ for $\\alpha \\ne 1$ and $\\Phi(t) = -\\frac{2}{\\pi} \\log \\abs{t}$ for $\\alpha = 1$, where $\\beta \\in [-1, 1]$, $\\gamma \u003e 0$ and $\\mu \\in \\R$.\nExample: let $X$ be [[normal|05AQE0QR0D]] with mean $0$ and variance $1$.\nThen $a X_1 + b X_2$ is normal with mean $0$ and variance $a^2 + b^2$.\nSo it has the law of $c X$ with $c = \\sqrt{a^2 + b^2}$, and the index is $2$.\nExample: for the standard [[Cauchy|0GPG24Q3GR]] law, $\\varphi(t) = e^{-\\abs{t}}$, so $a X_1 + b X_2$ has characteristic function $e^{-(a + b)\\abs{t}}$, the law of $(a + b) X$; this is index $1$.\nNon-example: let $X$ be Poisson with mean $\\lambda \u003e 0$. Then $X_1 + X_2$ is Poisson with mean $2\\lambda$, and takes all values in $\\N$. If it had the law of $cX + d$, comparing the two smallest values of the supports would force $c = 1$ and $d = 0$, but $X$ has mean $\\lambda \\ne 2\\lambda$.\nFor $\\alpha \u003c 2$, $E[\\abs{X}^p] \u003c \\infty$ if and only if $p \u003c \\alpha$; so the variance is infinite, and the mean is finite only for $\\alpha \u003e 1$.\nSome authors also call the point masses stable.\n",
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