Lévy distribution

Lévy (unshifted)
	Probability density function; ;
	Cumulative distribution function; ;
parameters:
support:
pdf:
cdf:
mean:	infinite
median:
mode:
variance:	infinite
skewness:	undefined
kurtosis:	undefined
entropy:
mgf:	undefined
cf:

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For the more general family of Lévy alpha-stable distributions, of which this distribution is a special case, see stable distribution.

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 stable distributions, which does not generally have an analytically expressible probability density.

[edit] Definition

The probability density function of the Lévy distribution over the domain $x\ge 0$ is

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

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

$F(x;c)=\textrm{erfc}\left(\sqrt{c/2x}\right)$

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

$\varphi(t;c)=e^{i\mu t-\sqrt{-2ict}}.$

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

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

The nth moment of the unshifted Lévy distribution is formally defined by:

$m_n\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{c}{2\pi}}\int_0^\infty \frac{e^{-c/2x}\,x^n}{x^{3/2}}\,dx$

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:

$M(t;c)\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{c}{2\pi}}\int_0^\infty \frac{e^{-c/2x+tx}}{x^{3/2}}\,dx$

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 probability density function exhibits heavy tail behavior falling off as:

$\lim_{x\rightarrow \infty}f(x;c) =\sqrt{\frac{c}{2\pi}}~\frac{1}{x^{3/2}}.$

This is illustrated in the diagram below, in which the probability density functions for various values of c are plotted on a log-log scale.

Probability density function for the Lévy distribution

[edit] Related distributions

Relation to stable distribution: If $X \sim \textrm{Levy}(c)$ then $X \sim \textrm{Stable}(1/2,1,c,0) \,$
Relation to Scale-inverse-chi-square distribution: If $X \sim \textrm{Levy}(c)$ then $X \sim \textrm{Scale-inv-}\chi^2(1,c)$
Relation to inverse gamma distribution: If $X \sim \textrm{Levy}(c)$ then $X \sim \textrm{Inv-Gamma}(1/2,c/2)$

[edit] 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).^[1]
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.

The length of the path followed by a photon in a turbid medium follows the Lévy distribution. ^[2]

The Lévy distribution has been used post 1987 crash by the Options Clearing Corporation for setting margin requirements because its parameters are more robust to extreme events than those of a normal distribution, and thus extreme events do not suddenly increase margin requirements which may worsen a crisis.^[3]

The statistics of solar flares are described by a non-Gaussian distribution. The solar flare statistics were shown to be describable by a Lévy distribution and it was assumed that intermittent solar flares perturb the intrinsic fluctuations in Earth’s average temperature. The end result of this perturbation is that the statistics of the temperature anomalies inherit the statistical structure that was evident in the intermittency of the solar flare data. ^[4]

[edit] Notes

^ "The Lévy distribution as maximizing one's chances of finding a tasty snack". http://www.livescience.com/animalworld/070403_fly_tricks.html. Retrieved April 7 2007.
^ Rogers, Geoffrey L, Multiple path analysis of reflectance from turbid media. Journal of the Optical Society of America A, 25:11, p 2879-2883 (2008).
^ Do economists make markets?: on the performativity of economics by Donald A. MacKenzie, Fabian Muniesa, Lucia Siu, Princeton University Press, 2007, ISBN 978 0 69113016 3, p. 80
^ Scafetta, N., Bruce, J.W., Is climate sensitive to solar variability? Physics Today, 60, 50-51 (2008) [1].

[edit] References

"Information on stable distributions". http://academic2.american.edu/~jpnolan/stable/stable.html. Retrieved July 13 2005. - 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 An introduction to stable distributions, Chapter 1

[0] "The Lévy distribution as maximizing one's chances of finding a tasty snack". http://www.livescience.com/animalworld/070403_fly_tricks.html. Retrieved April 7 2007.

[1] Rogers, Geoffrey L, Multiple path analysis of reflectance from turbid media. Journal of the Optical Society of America A, 25:11, p 2879-2883 (2008).

[2] Do economists make markets?: on the performativity of economics by Donald A. MacKenzie, Fabian Muniesa, Lucia Siu, Princeton University Press, 2007, ISBN 978 0 69113016 3, p. 80

[3] Scafetta, N., Bruce, J.W., Is climate sensitive to solar variability? Physics Today, 60, 50-51 (2008) [1].

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Lévy distribution

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Related distributions

[edit] Applications

[edit] Notes

[edit] References

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Probability density function
Cumulative distribution function
parameters:	$c > 0\,$
support:	$x \in [0, \infty)$
pdf:	$\sqrt{\frac{c}{2\pi}}~ \frac{e^{-c/2x}}{x^{3/2}}$
cdf:	$\textrm{erfc}\left(\sqrt{c/2x}\right)$
mean:	infinite
median:	$c/2(\textrm{erf}^{-1}(1/2))^2\,$
mode:	$\frac{c}{3}$
variance:	infinite
skewness:	undefined
kurtosis:	undefined
entropy:	$\frac{1+3\gamma+\ln(16\pi c^2)}{2}$
mgf:	undefined
cf:	$e^{-\sqrt{-2ict}}$