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. It is a stable distribution.

Definition
The probability density function of the Lévy distribution over the domain $$x \ge \mu$$ is


 * $$f(x; \mu, c) = \sqrt{\frac{c}{2\pi}} \, \frac{e^{-\frac{c}{2(x - \mu)}}}{(x - \mu)^{3/2}},$$

where $$\mu$$ is the location parameter, and $$c$$ is the scale parameter. The cumulative distribution function is


 * $$F(x; \mu, c) = \operatorname{erfc}\left(\sqrt{\frac{c}{2(x - \mu)}}\right) = 2 - 2 \Phi\left({\sqrt{\frac{c}{(x - \mu)}}}\right),$$

where $$\operatorname{erfc}(z)$$ 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:


 * $$f(x; \mu, c) \,dx = f(y; 0, 1) \,dy,$$

where y is defined as


 * $$y = \frac{x - \mu}{c}.$$

The characteristic function of the Lévy distribution is given by


 * $$\varphi(t; \mu, c) = e^{i\mu t - \sqrt{-2ict}}.$$

Note that the characteristic function can also be written in the same form used for the stable distribution with $$\alpha = 1/2$$ and $$\beta = 1$$:


 * $$\varphi(t; \mu, c) = e^{i\mu t - |ct|^{1/2} (1 - i\operatorname{sign}(t))}.$$

Assuming $$\mu = 0$$, the nth moment of the unshifted Lévy distribution is formally defined by


 * $$m_n\ \stackrel{\text{def}}{=}\ \sqrt{\frac{c}{2\pi}} \int_0^\infty \frac{e^{-c/2x} x^n}{x^{3/2}} \,dx,$$

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


 * $$M(t; c)\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{c}{2\pi}} \int_0^\infty \frac{e^{-c/2x + tx}}{x^{3/2}} \,dx,$$

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:


 * $$f(x; \mu, c) \sim \sqrt{\frac{c}{2\pi}} \, \frac{1}{x^{3/2}}$$ as $$x \to \infty,$$

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:


 * [[Image:Levy0 LdistributionPDF.svg|325px|thumb|left|Probability density function for the Lévy distribution on a log–log plot]]

The standard Lévy distribution satisfies the condition of being stable:


 * $$(X_1 + X_2 + \dotsb + X_n) \sim n^{1/\alpha}X,$$

where $$X_1, X_2, \ldots, X_n, X$$ are independent standard Lévy-variables with $$\alpha = 1/2.$$

Related distributions

 * If $$X \sim \operatorname{Levy}(\mu, c)$$, then $$kX + b \sim \operatorname{Levy}(k\mu + b, kc).$$
 * If $$X \sim \operatorname{Levy}(0, c)$$, then $$X \sim \operatorname{Inv-Gamma}(1/2, c/2)$$ (inverse gamma distribution). Here, the Lévy distribution is a special case of a Pearson type V distribution.
 * If $$Y \sim \operatorname{Normal}(\mu, \sigma^2)$$ (normal distribution), then $$(Y - \mu)^{-2} \sim \operatorname{Levy}(0, 1/\sigma^2).$$
 * If $$X \sim \operatorname{Normal}(\mu, 1/\sqrt{\sigma})$$, then $$(X - \mu)^{-2} \sim \operatorname{Levy}(0, \sigma)$$.
 * If $$X \sim \operatorname{Levy}(\mu, c)$$, then $$X \sim \operatorname{Stable}(1/2, 1, c, \mu)$$ (stable distribution).
 * If $$X \sim \operatorname{Levy}(0, c)$$, then $$X\,\sim\,\operatorname{Scale-inv-\chi^2}(1, c)$$ (scaled-inverse-chi-squared distribution).
 * If $$X \sim \operatorname{Levy}(\mu, c)$$, then $$(X - \mu)^{-1/2} \sim \operatorname{FoldedNormal}(0, 1/\sqrt{c})$$ (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


 * $$X = F^{-1}(U) = \frac{c}{(\Phi^{-1}(1 - U/2))^2} + \mu$$

is Lévy-distributed with location $$\mu$$ and scale $$c$$. Here $$\Phi(x)$$ 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 $$\alpha$$ from the starting point, by the Brownian motion has the Lévy distribution with $$c=\alpha^2$$. (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.