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. It is a stable distribution.

Definition

The probability density function of the Lévy distribution over the domain x \ge \mu is where \mu is the location parameter, and c is the scale parameter. The cumulative distribution function is where is the complementary error function, and \Phi(x) is the Laplace function (CDF of the standard normal distribution). The shift parameter \mu has the effect of shifting the curve to the right by an amount \mu and changing the support to the interval [\mu, \infty). Like all stable distributions, the Lévy 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 \beta = 1: Assuming \mu = 0, the nth moment of the unshifted Lévy distribution is formally defined by which diverges for all n \geq 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 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: 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 \mu = 0 are plotted on a log–log plot: The standard Lévy distribution satisfies the condition of being stable: where are independent standard Lévy-variables with

Related distributions

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 \mu and scale c. Here \Phi(x) is the cumulative distribution function of the standard normal distribution.

Applications

Footnotes