---
tag: Q82B4H4Z4J
kind: statement
role: theorem
name: Lévy–Khintchine representation
msc: ["60E07", 60G51, "60E10"]
uses: [BX3V3NYM1Y, 0V5B6D6BQH, 059TJN0NZ1, 8NT5CFG2ZR]
about: [BX3V3NYM1Y, 8NT5CFG2ZR]
sources:
  - url: https://en.wikipedia.org/wiki/L%C3%A9vy_process
---

## Statement

Let $\mu$ be a [[probability measure|059TJN0NZ1]] on $\R$ with [[characteristic function|0V5B6D6BQH]] $\varphi(u) = \int_\R e^{iux} \, \mu(dx)$.
Then $\mu$ 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.
The measure satisfies $\int_{\R \setminus \set{0}} \min(1, x^2) \, \nu(dx) < \infty$, and $\varphi(u) = e^{\psi(u)}$ for all $u \in \R$, where
$$\psi(u) = i b u - \frac{\sigma^2 u^2}{2} + \int_{\R \setminus \set{0}} \left(e^{iux} - 1 - iux \, \mathbf{1}_{\set{\abs{x} < 1}}\right) \nu(dx).$$
In that case the triple $(b, \sigma^2, \nu)$ is uniquely determined by $\mu$.

## Remarks

The 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 $\mathbf{1}_{\set{\abs{x} < 1}}$.
The three parts are a drift, a Gaussian part with variance $\sigma^2$, and a jump part.
Proof idea for "only if": write $\mu$ as the $n$-fold convolution power of some $\mu_n$.
Then $\exp(n(\varphi_n - 1))$ is the characteristic function of a compound Poisson law, and it tends to $\varphi$.
The triple is obtained as a limit of the jump measures $n \mu_n$.
Example: the [[normal|05AQE0QR0D]] law with mean $m$ and variance $\sigma^2$ has the triple $(m, \sigma^2, 0)$.
Example: 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 $\psi(u) = \lambda(e^{iu} - 1)$.
For 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 $t \ge 0$.
The same theorem holds on $\R^d$, with $\sigma^2$ replaced by a positive semidefinite matrix and $ux$ by a dot product.

## Explanations

### L1 human

Every 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.
The theorem says this precisely through the characteristic function, and the three ingredients are unique.
The measure $\nu$ tells how often jumps of each size happen.

### L2 human

The theorem classifies [[infinitely divisible|BX3V3NYM1Y]] laws, and through them all [[Lévy processes|8NT5CFG2ZR]].
The Lévy–Itô decomposition writes a Lévy process as a Brownian motion with drift plus an independent jump process.
The jump process is built from a Poisson random measure with intensity $\nu$.
[[Stable laws|F3N6E1Y2AY]] with index $\alpha \in (0, 2)$ are the case $\sigma = 0$ and $\nu(dx) = (c_+ \mathbf{1}_{\set{x > 0}} + c_- \mathbf{1}_{\set{x < 0}}) \abs{x}^{-1-\alpha} \, dx$ with $c_+, c_- \ge 0$ not both $0$.
The condition $\int \min(1, x^2) \, d\nu < \infty$ allows infinitely many small jumps, as long as they are square-summable.