Parameters Probability density function${\displaystyle \mu =0,\;c=\pi /2}$ ${\displaystyle c\in (0,\infty )}$ — 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, 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:

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

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.

## Properties

• 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.

### Approximations

In the "standard" case ${\displaystyle \mu =0}$ and ${\displaystyle c=\pi /2}$, the pdf can be approximated[4] using Lindhard theory which says:

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

where ${\displaystyle \gamma }$ is Euler's constant.

A similar approximation [5] of ${\displaystyle p(x;\mu ,c)}$ for ${\displaystyle \mu =0}$ and ${\displaystyle c=1}$ is:

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

## Related distributions

• The Landau distribution is a stable distribution with stability parameter ${\displaystyle \alpha }$ 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. ^
5. ^ Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).