Landau distribution

In probability theory, the Landau distribution is a probability distribution named after Lev Landau. Because of the distribution's "fat" tail, the moments of the distribution, such as mean or variance, are undefined. The distribution is a particular case of stable distribution.

Definition

The probability density function, as written originally by Landau, is defined by the complex integral:

$p(x)=\frac{1}{2\pi i}\underset{a-i\mathrm{\infty }}{\overset{a+i\mathrm{\infty }}{\int }}{e}^{s\log (s)+xs}ds,$

where a is an arbitrary positive real number, meaning that the integration path can be any parallel to the imaginary axis, intersecting the real positive semi-axis, and $\log$ refers to the natural logarithm. In other words it is the Laplace transform of the function $s^s$.

The following real integral is equivalent to the above:

$p(x)=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t\log (t)-xt}\sin (\pi t)dt.$

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters $\alpha =1$ and $\beta =1$, with characteristic function:

$\phi (t;\mu ,c)=\exp (it\mu -\frac{2ict}{\pi }\log |t|-c|t|)$

where $c\in (0,\mathrm{\infty })$ and $\mu \in (-\mathrm{\infty },\mathrm{\infty })$, which yields a density function:

$p(x;\mu ,c)=\frac{1}{\pi c}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}\cos (t\left(\frac{x-\mu }{c}\right)+\frac{2t}{\pi }\log \left(\frac{t}{c}\right))dt,$

Taking $\mu =0$ and $c=\frac{\pi }{2}$ we get the original form of $p(x)$ above.

Properties

The approximation function for $\mu =0,c=1$

• Translation: If $X\sim \text{Landau}(\mu ,c)$ then $X+m\sim \text{Landau}(\mu +m,c)$.
• Scaling: If $X\sim \text{Landau}(\mu ,c)$ then $aX\sim \text{Landau}(a\mu -\frac{2ac\log (a)}{\pi },ac)$.
• Sum: If $X\sim \text{Landau}({\mu }_1,c_1)$ and $Y\sim \text{Landau}({\mu }_2,c_2)$ then $X+Y\sim \text{Landau}({\mu }_1+{\mu }_2,c_1+c_2)$.

These properties can all be derived from the characteristic function. Together they imply that the Landau distributions are closed under affine transformations.

Approximations

In the "standard" case $\mu =0$ and $c=\pi /2$, the pdf can be approximated using Lindhard theory which says:

$p(x+\log (x)-1+\gamma )\approx \frac{\exp (-1/x)}{x(1+x)},$

where $\gamma$ is Euler's constant.

A similar approximation of $p(x;\mu ,c)$ for $\mu =0$ and $c=1$ is:

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

Related distributions

• The Landau distribution is a stable distribution with stability parameter $\alpha$ and skewness parameter $\beta$ both equal to 1.