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# Landau distribution

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

In probability theory, the Landau distribution[1] is a probability distribution named after Lev Landau. Because of the distribution's "fat" tail, the moments of the distribution, like 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:

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

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 ${\displaystyle c=1}$:

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

## Related distributions

• 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.

## References

1. ^ Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR). 8: 201.
2. ^ Gentle, James E. (2003). Random Number Generation and Monte Carlo Methods. Statistics and Computing (2nd ed.). New York, NY: Springer. p. 196. doi:10.1007/b97336. ISBN 978-0-387-00178-4.
3. ^ Zolotarev, V.M. (1986). One-dimensional stable distributions. Providence, R.I.: American Mathematical Society. ISBN 0-8218-4519-5.
4. ^ Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).