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.
The probability density function, as written originally by Landau, is defined by the complex integral:
p(x)=
| 1 | |
| 2\pii |
| a+iinfty | |
| \int | |
| a-iinfty |
esds,
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
ss
The following real integral is equivalent to the above:
p(x)=
| 1 | |
| \pi |
| infty | |
| \int | |
| 0 |
e-t\sin(\pit)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
\beta=1
\varphi(t;\mu,c)=\exp\left(it\mu-\tfrac{2ict}{\pi}log|t|-c|t|\right)
where
c\in(0,infty)
\mu\in(-infty,infty)
p(x;\mu,c)=
| 1 | |
| \pic |
| infty | |
| \int | |
| 0 |
e-t\cos\left(t\left(
| x-\mu | \right)+ | |
| c |
| 2t | log\left( | |
| \pi |
| t | |
| c |
\right)\right)dt,
Taking
\mu=0
| c= | \pi |
| 2 |
p(x)
X\simrm{Landau}(\mu,c)
X+m\simrm{Landau}(\mu+m,c)
X\simrm{Landau}(\mu,c)
aX\simrm{Landau}(a\mu-\tfrac{2aclog(a)}{\pi},ac)
X\simrm{Landau}(\mu1,c1)
Y\simrm{Landau}(\mu2,c2)
X+Y\simrm{Landau}(\mu1+\mu2,c1+c2)
These properties can all be derived from the characteristic function.Together they imply that the Landau distributions are closed under affine transformations.
In the "standard" case
\mu=0
c=\pi/2
p(x+log(x)-1+\gamma) ≈
| \exp(-1/x) | |
| x(1+x) |
,
where
\gamma
A similar approximation [5] of
p(x;\mu,c)
\mu=0
c=1
p(x) ≈
| 1 | |
| \sqrt{2\pi |
\alpha
\beta