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, 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:


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

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:


 * $$p(x) = \frac{1}{\pi} \int_0^\infty e^{-t \log(t) - x t} \sin(\pi t)\, dt.$$

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.