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

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

• 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.
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).