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 is

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

is the complementary error function is the Laplace function (CDF of the standard normal distribution,). 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

:

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

, 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

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 are plotted on a log–log plot:

The standard Lévy distribution satisfies the condition of being stable:

Related distributions

• (inverse gamma distribution). Here, the Lévy distribution is a special case of a Pearson type V distribution.

• (normal distribution

• .

• (stable distribution).

• (scaled-inverse-chi-squared distribution).

• (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

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 from the starting point, by the Brownian motion. (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.

• A Cauchy process can be defined as a Brownian motion subordinated to a process associated with a Lévy distribution.

Footnotes

Notes

References

• - 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