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  "body": "\n## Definition\n\nLet $\\mu$ be a [[probability distribution|0DVDCMA2DK]] on $\\R$.\n$\\mu$ is *infinitely divisible* if for every integer $n \\ge 1$ there are [[independent and identically distributed|00MRN0FARV]] random variables $Y_{n,1}, \\dots, Y_{n,n}$ such that $Y_{n,1} + \\dots + Y_{n,n}$ has law $\\mu$.\nA real random variable is infinitely divisible if its law is.\n\n## Remarks\n\nEquivalently, for every $n \\ge 1$ there is a probability measure $\\mu_n$ whose $n$-fold [[convolution|0F8GDKM0A7]] power is $\\mu$; or, in terms of [[characteristic functions|0V5B6D6BQH]], $\\varphi = \\varphi_n^n$ for some characteristic function $\\varphi_n$.\nThe same definition applies to laws on $\\R^d$.\nExample: the Poisson law with mean $\\lambda$ is the law of a sum of $n$ independent Poisson variables with mean $\\lambda / n$, since $e^{\\lambda(e^{it} - 1)} = \\left(e^{(\\lambda/n)(e^{it} - 1)}\\right)^n$.\nExample: the normal law with mean $m$ and variance $\\sigma^2$ is the law of a sum of $n$ independent normal variables with mean $m/n$ and variance $\\sigma^2/n$.\nNon-example: a [[Bernoulli|0386Y0Z7Y7]] law with parameter $p \\in (0, 1)$.\nSuppose $Y_1 + Y_2$ has this law, with $Y_1, Y_2$ i.i.d.\nIf $s \u003c s'$ were two points of the support of $Y_1$, then $2s \u003c s + s' \u003c 2s'$ would be three values in $\\set{0, 1}$.\nSo $Y_1$ is constant, and then $Y_1 + Y_2$ is constant, a contradiction.\nThe characteristic function of an infinitely divisible law has no zeros.\n",
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