Landau distribution

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

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

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 ${\displaystyle \log }$ refers to the natural logarithm.

The following real integral is equivalent to the above:

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

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters ${\displaystyle \alpha =1}$ and ${\displaystyle \beta =1}$,[2] with characteristic function:[3]

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

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

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

Let us note that the original form of ${\displaystyle p(x)}$ is obtained for ${\displaystyle \mu =0}$ and ${\displaystyle c={\frac {\pi }{2}}}$, while the following is an approximation[4] of ${\displaystyle p(x;\mu ,c)}$ for ${\displaystyle \mu =0}$ and ${\displaystyle c=1}$:

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

• If ${\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,}$ then ${\displaystyle X+m\sim {\textrm {Landau}}(\mu +m,c)\,}$.
• The Landau distribution is a stable distribution with stability parameter ${\displaystyle \alpha }$ and skewness parameter ${\displaystyle \beta }$ both equal to 1.