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  "body": "\n## Statement\n\nLet $\\mu$ be a [[probability measure|059TJN0NZ1]] on $\\R$ with [[characteristic function|0V5B6D6BQH]] $\\varphi(u) = \\int_\\R e^{iux} \\, \\mu(dx)$.\nThen $\\mu$ is [[infinitely divisible|BX3V3NYM1Y]] if and only if there are $b \\in \\R$, $\\sigma \\ge 0$ and a measure $\\nu$ on $\\R \\setminus \\set{0}$ with the following properties.\nThe measure satisfies $\\int_{\\R \\setminus \\set{0}} \\min(1, x^2) \\, \\nu(dx) \u003c \\infty$, and $\\varphi(u) = e^{\\psi(u)}$ for all $u \\in \\R$, where\n$$\\psi(u) = i b u - \\frac{\\sigma^2 u^2}{2} + \\int_{\\R \\setminus \\set{0}} \\left(e^{iux} - 1 - iux \\, \\mathbf{1}_{\\set{\\abs{x} \u003c 1}}\\right) \\nu(dx).$$\nIn that case the triple $(b, \\sigma^2, \\nu)$ is uniquely determined by $\\mu$.\n\n## Remarks\n\nThe measure $\\nu$ is the Lévy measure and $(b, \\sigma^2, \\nu)$ is the characteristic triple; the value of $b$ depends on the choice of the cut-off $\\mathbf{1}_{\\set{\\abs{x} \u003c 1}}$.\nThe three parts are a drift, a Gaussian part with variance $\\sigma^2$, and a jump part.\nProof idea for \"only if\": write $\\mu$ as the $n$-fold convolution power of some $\\mu_n$.\nThen $\\exp(n(\\varphi_n - 1))$ is the characteristic function of a compound Poisson law, and it tends to $\\varphi$.\nThe triple is obtained as a limit of the jump measures $n \\mu_n$.\nExample: the [[normal|05AQE0QR0D]] law with mean $m$ and variance $\\sigma^2$ has the triple $(m, \\sigma^2, 0)$.\nExample: the [[Poisson|0XB9GQTN9G]] law with mean $\\lambda$ has the triple $(0, 0, \\lambda \\delta_1)$, since the cut-off term vanishes at $x = 1$ and $\\psi(u) = \\lambda(e^{iu} - 1)$.\nFor a [[Lévy process|8NT5CFG2ZR]] $(X_t)_{t \\ge 0}$, the law of $X_1$ is infinitely divisible with exponent $\\psi$, and $E[e^{iuX_t}] = e^{t \\psi(u)}$ for all $t \\ge 0$.\nThe same theorem holds on $\\R^d$, with $\\sigma^2$ replaced by a positive semidefinite matrix and $ux$ by a dot product.\n\n## Explanations\n\n### L1 human\n\nEvery law that can be split into any number of equal independent parts is built from three ingredients: a steady drift, a normal fluctuation, and random jumps of various sizes.\nThe theorem says this precisely through the characteristic function, and the three ingredients are unique.\nThe measure $\\nu$ tells how often jumps of each size happen.\n\n### L2 human\n\nThe theorem classifies [[infinitely divisible|BX3V3NYM1Y]] laws, and through them all [[Lévy processes|8NT5CFG2ZR]].\nThe Lévy–Itô decomposition writes a Lévy process as a Brownian motion with drift plus an independent jump process.\nThe jump process is built from a Poisson random measure with intensity $\\nu$.\n[[Stable laws|F3N6E1Y2AY]] with index $\\alpha \\in (0, 2)$ are the case $\\sigma = 0$ and $\\nu(dx) = (c_+ \\mathbf{1}_{\\set{x \u003e 0}} + c_- \\mathbf{1}_{\\set{x \u003c 0}}) \\abs{x}^{-1-\\alpha} \\, dx$ with $c_+, c_- \\ge 0$ not both $0$.\nThe condition $\\int \\min(1, x^2) \\, d\\nu \u003c \\infty$ allows infinitely many small jumps, as long as they are square-summable.\n",
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