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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.

Lévy (unshifted)
Probability density function
Levy distribution PDF
Cumulative distribution function
Levy distribution CDF
Parameters location; scale
Support
PDF
CDF
Mean
Median, for
Mode, for
Variance
Skewnessundefined
Ex. kurtosisundefined
Entropy where is Euler's constant
MGFundefined
CF


Contents

DefinitionEdit

The probability density function of the Lévy distribution over the domain   is

 

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

 

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

 

where y is defined as

 

The characteristic function of the Lévy distribution is given by

 

Note that the characteristic function can also be written in the same form used for the stable distribution with   and  :

 

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

 

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

 

which diverges for   and is therefore not defined in an interval around zero, so that the moment generating function is not defined per se. 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:

    as    

(This shows that Lévy 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   are plotted on a log-log scale.

 
Probability density function for the Lévy distribution on a log-log scale.


The standard Lévy distribution satisfies the condition of being Stable

 ,

where   are independent standard Lévy-variables with  .

Related distributionsEdit

  • If   then  
  • If   then   (inverse gamma distribution)
  • Lévy distribution is a special case of type 5 Pearson distribution
  • If   (Normal distribution) then  
  • If   then  
  • If   then   (Stable distribution)
  • If   then   (Scaled-inverse-chi-squared distribution)
  • If   then   (Folded normal distribution)

Random sample generationEdit

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]

 

is Lévy-distributed with location   and scale  . Here   is the cumulative distribution function of the standard normal distribution.

ApplicationsEdit

FootnotesEdit

  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]

NotesEdit

  1. ^ How to derive the function for a random sample from a Lévy Distribution: 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.
  3. ^ Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.

ReferencesEdit

External linksEdit