Lévy distribution

Short description: Probability distribution

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

Lévy (unshifted)

Probability density function

Cumulative distribution function

Parameters|  μ location; c>0 scale
Support|  x∈(μ,∞)
PDF|  c2πe−c2(x−μ)(x−μ)3/2
CDF|  erfc(c2(x−μ))
Quantile|  μ+σ2(erfc−1(p))2
Mean|  ∞
Median|  μ+c/2(erfc−1(1/2))2
Mode|  μ+c3
Variance|  ∞
Skewness|  undefined
Kurtosis|  undefined
Entropy|

1+3γ+ln⁡(16πc2)2

where γ is the Euler-Mascheroni constant
MGF|  undefined
CF|  eiμt−−2ict

In probability theory and statistics, the Lévy distribution, named after Paul 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.[note 1] It is a special case of the inverse-gamma distribution. It is a stable distribution.

## Contents

* 1 Definition
* 2 Related distributions
* 3 Random-sample generation
* 4 Applications
* 5 Footnotes
* 6 Notes
* 7 References
* 8 External links

## Definition

The probability density function of the Lévy distribution over the domain x≥μ is

f(x;μ,c)=c2πe−c2(x−μ)(x−μ)3/2,

where μ is the location parameter, and c is the scale parameter. The cumulative distribution function is

F(x;μ,c)=erfc⁡(c2(x−μ))=2−2Φ(c(x−μ)),

where erfc⁡(z) is the complementary error function, and Φ(x) is the Laplace function (CDF of the standard normal distribution). The shift parameter μ has the effect of shifting the curve to the right by an amount μ and changing the support to the interval [μ, ∞). Like all stable distributions, the Lévy distribution has a standard form f(x; 0, 1) which has the following property:

f(x;μ,c)dx=f(y;0,1)dy,

where y is defined as

y=x−μc.

The characteristic function of the Lévy distribution is given by

φ(t;μ,c)=eiμt−−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:

φ(t;μ,c)=eiμt−|ct|1/2(1−isign⁡(t)).

Assuming μ=0, the nth moment of the unshifted Lévy distribution is formally defined by

mn =def c2π∫0∞e−c/2xxnx3/2dx,

which diverges for all n≥1/2, so that the integer moments of the Lévy distribution do not exist (only some fractional moments).

The moment-generating function would be formally defined by

M(t;c) =def c2π∫0∞e−c/2x+txx3/2dx,

however, this diverges for t>0 and is therefore not defined on an interval around zero, so the moment-generating function is actually undefined.

Like all stable distributions except the normal distribution, the wing of the probability density function exhibits heavy tail behavior falling off according to a power law:

f(x;μ,c)∼c2π1x3/2 as x->∞,

which shows that the Lévy distribution is not just heavy-tailed but also fat-tailed. This is illustrated in the diagram below, in which the probability density functions for various values of c and μ=0 are plotted on a log–log plot:

Probability density function for the Lévy distribution on a log–log plot

The standard Lévy distribution satisfies the condition of being stable:

(X1+X2+⋯+Xn)∼n1/αX,

where X1,X2,…,Xn,X are independent standard Lévy-variables with α=1/2.

## Related distributions

* If X∼Levy⁡(μ,c), then kX+b∼Levy⁡(kμ+b,kc).
* If X∼Levy⁡(0,c), then X∼Inv-Gamma⁡(1/2,c/2) (inverse gamma distribution). Here, the Lévy distribution is a special case of a Pearson type V distribution.
* If Y∼Normal⁡(μ,σ2) (normal distribution), then (Y−μ)−2∼Levy⁡(0,1/σ2).
* If X∼Normal⁡(μ,1/σ), then (X−μ)−2∼Levy⁡(0,σ).
* If X∼Levy⁡(μ,c), then X∼Stable⁡(1/2,1,c,μ) (stable distribution).
* If X∼Levy⁡(0,c), then X∼Scale−inv−χ2⁡(1,c) (scaled-inverse-chi-squared distribution).
* If X∼Levy⁡(μ,c), then (X−μ)−1/2∼FoldedNormal⁡(0,1/c) (folded normal distribution).

## Random-sample generation

Random samples from the Lévy distribution can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0, 1], the variate X given by[1]

X=F−1(U)=c(Φ−1(1−U/2))2+μ

is Lévy-distributed with location μ and scale c. Here Φ(x) is the cumulative distribution function of the standard normal distribution.

## Applications

* The frequency of geomagnetic reversals appears to follow a Lévy distribution
* The time of hitting a single point, at distance α from the starting point, by the Brownian motion has the Lévy distribution with c=α2. (For a Brownian motion with drift, this time may follow an inverse Gaussian distribution, which has the Lévy distribution as a limit.)
* The length of the path followed by a photon in a turbid medium follows the Lévy distribution.[2]
* A Cauchy process can be defined as a Brownian motion subordinated to a process associated with a Lévy distribution.[3]

## Footnotes

1. ↑ "van der Waals profile" appears with lowercase "van" in almost all sources, such as: Statistical mechanics of the liquid surface by Clive Anthony Croxton, 1980, A Wiley-Interscience publication, ISBN 0-471-27663-4, ISBN 978-0-471-27663-0, [1]; and in Journal of technical physics, Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995, [2]

## Notes

1. ↑ "The Lévy Distribution". The University of Alabama in Huntsville, Department of Mathematical Sciences. http://www.math.uah.edu/stat/special/Levy.html.
2. ↑ Rogers, Geoffrey L. (2008). "Multiple path analysis of reflectance from turbid media". Journal of the Optical Society of America A 25 (11): 2879–2883. doi:10.1364/josaa.25.002879. PMID 18978870. Bibcode: 2008JOSAA..25.2879R.
3. ↑ Applebaum, D.. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes". University of Sheffield. pp. 37–53. http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf.

## References

* "Information on stable distributions". https://edspace.american.edu/jpnolan/stable/. \- 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

## External links

* Weisstein, Eric W.. "Lévy Distribution". http://mathworld.wolfram.com/LevyDistribution.html.

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