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. In other words it is the Laplace transform of the function ${\displaystyle s^{s}}$.

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:

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

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

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

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

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