Landau distribution

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:

:$p(x) = \frac \int_^ e^\, ds ,$

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:

:$p(x) = \frac \int_0^\infty e^ \sin(\pi t)\, dt.$

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:

:$\varphi(t;\mu,c)=\exp\left(it\mu -\tfrac\log, t, -c, t, \right)$

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

:$p(x;\mu,c) = \frac\int_^ e^\cos\left(t\left(\frac\right)+\frac\log\left(\frac\right)\right)\, dt ,$

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

Properties

* Translation: If $X \sim \textrm(\mu,c)\,$ then $X + m \sim \textrm(\mu + m ,c) \,$.
* Scaling: If $X \sim \textrm(\mu,c)\,$ then $aX \sim \textrm(a\mu-\tfrac, ac) \,$.
* Sum: If $X \sim \textrm(\mu_1, c_1)$ and $Y \sim \textrm(\mu_2, c_2) \,$ then $X+Y \sim \textrm(\mu_1+\mu_2, c_1+c_2)$.

These properties can all be derived from the characteristic function.
Together they imply that the Landau distribution is 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:

:$p(x+\log(x)-1+\gamma) \approx \frac,$

where $\gamma$ is Euler's constant.

A similar approximation of $p(x;\mu,c)$ for $\mu=0$ and $c=1$ is:

:$p(x) \approx \frac\exp\left(-\frac\right).$

Related distributions

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

References

{{DEFAULTSORT:Landau Distribution
Continuous distributions
Probability distributions with non-finite variance
Power laws
Stable distributions
Lev Landau