Lévy distribution

Probability distribution
name =Lévy (unshifted)
type =density
pdf_


cdf_


parameters =$c\; >\; 0,$
support =$x\; in\; [0,\; infty)$
pdf =$sqrt\{frac\{c\}\{2pi$~frac{e^{-c/2x{x^{3/2
cdf =$extrm\{erfc\}left(sqrt\{c/2x\}\; ight)$
mean =infinite
median =$c/2(\; extrm\{erf\}^\{-1\}(1/2))^2,$
mode =$frac\{c\}\{3\}$
variance =infinite
skewness =undefined
kurtosis =undefined
entropy =$frac\{1+3gamma+ln(16pi\; c^2)\}\{2\}$
mgf =undefined| char =$e^\{-sqrt\{-2ict$
In probability theory and statistics, the Lévy distribution, named after Paul Pierre Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy this distribution, with frequency as the dependent variable, is known as a Van der Waals profile.

It is one of the few distributions that are stable and that have probability density functions that are analytically expressible, the others being the normal distribution and the Cauchy distribution. All three are special cases of the Lévy skew alpha-stable distribution, which does not generally have an analytically expressible probability density.

The probability density function of the Lévy distribution over the domain $xge\; 0$ is

:$f(x;c)=sqrt\{frac\{c\}\{2pi~~frac\{e^\{-c/2x\{x^\{3/2$

where $c$ is the scale parameter. The cumulative distribution function is

:$F(x;c)=\; extrm\{erfc\}left(sqrt\{c/2x\}\; ight)$

where $extrm\{erfc\}(z)$ is the complementary error function. A shift parameter $mu$ may be included by replacing each occurrence of $x$ in the above equations with $x-mu$. This will simply have the effect of shifting the curve to the right by an amount $mu$, and changing the support to the interval [$mu$, $infty$). The characteristic function of the Lévy distribution (including a shift $mu$) is given by

:$varphi(t;c)=e^\{imu\; t-sqrt\{-2ict.$

Note that the characteristic function can also be written in the same form used for the Lévy skew alpha-stable distribution with $alpha=1/2$ and :

:$varphi(t;c)=e^\{imu\; t-|ct|^\{1/2\}~(1-i~\; extrm\{sign\}(t))\}.$

The "n"th moment of the unshifted Lévy distribution is formally defined by:

:$m\_n\; stackrel\{mathrm\{def\{=\}\; sqrt\{frac\{c\}\{2piint\_0^infty\; frac\{e^\{-c/2x\},x^n\}\{x^\{3/2,dx$

which diverges for all "n" > 0 so that the moments of the Lévy distribution do not exist. The moment generating function is formally defined by:

:$M(t;c)\; stackrel\{mathrm\{def\{=\}\; sqrt\{frac\{c\}\{2piint\_0^infty\; frac\{e^\{-c/2x+tx\{x^\{3/2,dx$

which diverges for $t>0$ and is therefore not defined in an interval around zero, so that the moment generating function is not defined "per se". In the wings of the distribution, the PDF exhibits heavy tail behavior falling off as:

:$lim\_\{x\; ightarrow\; infty\}f(x;c)\; =sqrt\{frac\{c\}\{2pi~frac\{1\}\{x^\{3/2.$

This is illustrated in the diagram below, in which the PDF's for various values of "c" are plotted on a log-log scale.

Related distributions

* Relation to Lévy skew alpha-stable distribution: If $X\; sim\; extrm\{Levy\}(c)$ then $X\; sim\; extrm\{Levy-S\}alpha\; extrm\{S\}(1/2,1,c,0)$
* Relation to Scale-inverse-chi-square distribution: If $X\; sim\; extrm\{Levy\}(c)$ then $X\; sim\; extrm\{Scale-inv-\}chi^2(1,c)$
* Relation to inverse gamma distribution: If $X\; sim\; extrm\{Levy\}(c)$ then $X\; sim\; extrm\{Inv-Gamma\}(1/2,c/2)$

Applications

*The Lévy distribution is of interest to the financial modeling community due to its empirical similarity to the returns of securities.
*It is claimed that fruit flies follow a form of the distribution to find food (Lévy flight). [cite web | title=The Lévy distribution as maximizing one's chances of finding a tasty snack| work= | url=http://www.livescience.com/animalworld/070403_fly_tricks.html | accessmonthday=April 7 | accessyear=2007]
* The frequency of geomagnetic reversals appears to follow a Lévy distribution
*The time of hitting a single point (different from the starting point 0) by the Brownian motion has the Lévy distribution.

References and external links

* - John P. Nolan's introduction to stable distributions, some papers on stable laws, and a free program to compute stable densities, cumulative distribution functions, quantiles, estimate parameters, etc. See especially [http://academic2.american.edu/~jpnolan/stable/chap1.pdf An introduction to stable distributions, Chapter 1]