Lévy distribution


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. It is a special case of the inverse-gamma distribution and a stable distribution.

Definition

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, and is the Laplace function. Like all stable distributions, the Lévy distribution has a standard form 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, so that the integer moments of the Lévy distribution do not exist.
The moment-generating function would be formally defined by
however, this diverges for 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:
as
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 are plotted on a log–log plot:
[file:Levy0 LdistributionPDF.svg|325px|thumb|center|Probability density function for the Lévy distribution on a log–log plot]
The standard Lévy distribution satisfies the condition of being stable:
where are independent standard Lévy-variables with

Related distributions

  • If, then
  • If, then . Here, the Lévy distribution is a special case of a Pearson type V distribution.
  • If, then
  • If, then.
  • If, then .
  • If, then .
  • If, then .

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

Applications