Landau distribution

Landau distribution

In probability theory, the Landau distribution is a probability distribution named after Lev Landau. Because of the distribution's long tail, the moments of the distribution, like mean or variance, are undefined. The distribution is a special case of the stable distribution.

Definition

The probability density function of a standard version of the Landau distribution is defined by the complex integral

$p(x) = \frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty}\! e^{s \log s + x s}\, ds ,$

where c is any positive real number, and log refers to the logarithm base e, the natural logarithm. The result does not change if c changes. For numerical purposes it is more convenient to use the following equivalent form of the integral,

$p(x) = \frac{1}{\pi} \int_0^\infty\! e^{-t \log t - x t} \sin(\pi t)\, dt.$

The full family of Landau distributions is obtained by extending the standard distribution to a location-scale family. This distribution can be approximated by

$p(x) = \frac{1}{\sqrt{2\pi}} \exp\left\{-\frac{1}{2}(x + e^{-x})\right\}.$

This distribution is a special case of the stable distribution with parameters α = 1, and β = 1.

The characteristic function may be expressed as:

$\varphi(t;\mu,c)=\exp\!\Big[\; it\mu - |c\,t|(1+\tfrac{2i}{\pi}\log(|t|))\Big].$

where μ and c are real, which yields a Landau distribution shifted by μ and scaled by c.

Related distributions

• If $X \sim \textrm{Landau}(\mu,c)\,$ then $X + m \sim \textrm{Landau}(\mu + m ,c) \,$
• The Landau distribution is a stable distribution