Landau distribution

In probability theory, the Landau distribution<ref>Template:Cite journal</ref> 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:

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

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 <math>\log</math> refers to the natural logarithm. In other words it is the Laplace transform of the function <math>s^s</math>.

The following real integral is equivalent to the above:

<math>p(x) = \frac{1}{\pi} \int_0^\infty e^{-t \log(t) - x t} \sin(\pi t)\, dt.</math>

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters <math>\alpha=1</math> and <math>\beta=1</math>,<ref>Template:Cite book </ref> with characteristic function:<ref>Template:Cite book</ref>

<math>\varphi(t;\mu,c)=\exp\left(it\mu -\tfrac{2ict}{\pi}\log|t|-c|t|\right)</math>

where <math>c\in(0,\infty)</math> and <math>\mu\in(-\infty,\infty)</math>, which yields a density function:

<math>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 , </math>

Taking <math>\mu=0</math> and <math>c=\frac{\pi}{2}</math> we get the original form of <math>p(x)</math> above.

Properties

File:Landau pdf.svg

The approximation function for <math>\mu=0,\,c=1</math>

• Translation: If <math>X \sim \textrm{Landau}(\mu,c)\, </math> then <math> X + m \sim \textrm{Landau}(\mu + m ,c) \,</math>.
• Scaling: If <math>X \sim \textrm{Landau}(\mu,c)\, </math> then <math> aX \sim \textrm{Landau}(a\mu-\tfrac{2ac\log(a)}{\pi}, ac) \,</math>.
• Sum: If <math>X \sim \textrm{Landau}(\mu_1, c_1)</math> and <math>Y \sim \textrm{Landau}(\mu_2, c_2) \,</math> then <math> X+Y \sim \textrm{Landau}(\mu_1+\mu_2, c_1+c_2)</math>.

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 <math>\mu=0</math> and <math>c=\pi/2</math>, the pdf can be approximated<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> using Lindhard theory which says:

<math>p(x+\log(x)-1+\gamma) \approx \frac{\exp(-1/x)}{x(1+x)},</math>

where <math>\gamma</math> is Euler's constant.

A similar approximation <ref>Template:Cite book</ref> of <math>p(x;\mu,c)</math> for <math>\mu=0</math> and <math>c=1</math> is:

<math>p(x) \approx \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x + e^{-x}}{2}\right).</math>

Related distributions

• The Landau distribution is a stable distribution with stability parameter <math>\alpha</math> and skewness parameter <math>\beta</math> both equal to 1.