Landau distribution

The probability density function, as written originally by Landau, is defined by the complex integral:

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

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 $\backslash log$ 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:

:$p(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash infty\; e^\{-t\; \backslash log(t)\; -\; x\; t\}\; \backslash sin(\backslash pi\; t)\backslash ,\; dt.$

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters $\backslash alpha=1$ and $\backslash beta=1$,{{ cite book | last = Gentle | first = James E. | title = Random Number Generation and Monte Carlo Methods | edition = 2nd | publisher = Springer | location = New York, NY | date = 2003 | series=Statistics and Computing | isbn =978-0-387-00178-4 | doi = 10.1007/b97336 |page=196}} with characteristic function:{{cite book|last1=Zolotarev|first1=V.M.|title=One-dimensional stable distributions|date=1986|publisher=American Mathematical Society|location=Providence, R.I.|isbn=0-8218-4519-5|url-access=registration|url=https://archive.org/details/onedimensionalst00zolo_0}}

:$\backslash varphi(t;\backslash mu,c)=\backslash exp\backslash left(it\backslash mu\; -\backslash tfrac\{2ict\}\{\backslash pi\}\backslash log|t|-c|t|\backslash right)$

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

:$p(x;\backslash mu,c)\; =\; \backslash frac\{1\}\{\backslash pi\; c\}\backslash int\_\{0\}^\{\backslash infty\}\; e^\{-t\}\backslash cos\backslash left(t\backslash left(\backslash frac\{x-\backslash mu\}\{c\}\backslash right)+\backslash frac\{2t\}\{\backslash pi\}\backslash log\backslash left(\backslash frac\{t\}\{c\}\backslash right)\backslash right)\backslash ,\; dt\; ,$

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

Image:Landau_pdf.svg

• Translation: If $X\; \backslash sim\; \backslash textrm\{Landau\}(\backslash mu,c)\backslash ,$ then $X\; +\; m\; \backslash sim\; \backslash textrm\{Landau\}(\backslash mu\; +\; m\; ,c)\; \backslash ,$.
• Scaling: If $X\; \backslash sim\; \backslash textrm\{Landau\}(\backslash mu,c)\backslash ,$ then $aX\; \backslash sim\; \backslash textrm\{Landau\}(a\backslash mu-\backslash tfrac\{2ac\backslash log(a)\}\{\backslash pi\},\; ac)\; \backslash ,$.
• Sum: If $X\; \backslash sim\; \backslash textrm\{Landau\}(\backslash mu\_1,\; c\_1)$ and $Y\; \backslash sim\; \backslash textrm\{Landau\}(\backslash mu\_2,\; c\_2)\; \backslash ,$ then $X+Y\; \backslash sim\; \backslash textrm\{Landau\}(\backslash mu\_1+\backslash mu\_2,\; c\_1+c\_2)$.

These properties can all be derived from the characteristic function.

Together they imply that the Landau distributions are closed under affine transformations.

In the "standard" case $\backslash mu=0$ and $c=\backslash pi/2$, the pdf can be approximated{{Cite web|url=https://reference.wolfram.com/language/ref/LandauDistribution.html|title=LandauDistribution—Wolfram Language Documentation}} using Lindhard theory which says:

:$p(x+\backslash log(x)-1+\backslash gamma)\; \backslash approx\; \backslash frac\{\backslash exp(-1/x)\}\{x(1+x)\},$

where $\backslash gamma$ is Euler's constant.

A similar approximation {{ cite book | last1 = Behrens | first1 = S. E. | last2 = Melissinos | first2 = A.C. | title = Univ. of Rochester Preprint UR-776 (1981) }} of $p(x;\backslash mu,c)$ for $\backslash mu=0$ and $c=1$ is:

:$p(x)\; \backslash approx\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\backslash exp\backslash left(-\backslash frac\{x\; +\; e^\{-x\}\}\{2\}\backslash right).$

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

{{Reflist}}

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Landau Distribution}}

Category:Continuous distributions

Category:Probability distributions with non-finite variance

Category:Power laws

Category:Stable distributions

Category:Lev Landau