Landau distribution

Landau distribution

Probability density function $\mu =0,\;c=\pi /2$
Parameters	$c\in (0,\infty )$ — scale parameter $\mu \in (-\infty ,\infty )$ — location parameter
Support	$\mathbb {R}$
PDF	${\frac {1}{\pi c}}\int _{0}^{\infty }e^{-t}\cos \left(t\left({\frac {x-\mu }{c}}\right)+{\frac {2t}{\pi }}\log \left({\frac {t}{c}}\right)\right)\,dt$
Mean	Undefined
Variance	Undefined
MGF	Undefined
CF	$\exp \left(it\mu -{\frac {2ict}{\pi }}\log |t|-c|t|\right)$

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:

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

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:

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

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:

$\varphi (t;\mu ,c)=\exp \left(it\mu -{\tfrac {2ict}{\pi }}\log |t|-c|t|\right)$

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

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

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 $X\sim {\textrm {Landau}}(\mu ,c)\,$ then $X+m\sim {\textrm {Landau}}(\mu +m,c)\,$.
• Scaling: If $X\sim {\textrm {Landau}}(\mu ,c)\,$ then $aX\sim {\textrm {Landau}}(a\mu -{\tfrac {2ac\log(a)}{\pi }},ac)\,$.
• Sum: If $X\sim {\textrm {Landau}}(\mu _{1},c_{1})$ and $Y\sim {\textrm {Landau}}(\mu _{2},c_{2})\,$ then $X+Y\sim {\textrm {Landau}}(\mu _{1}+\mu _{2},c_{1}+c_{2})$.

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:

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

Related distributions

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