Lévy distribution

For the more general family of Lévy alpha-stable distributions, of which this distribution is a special case, see stable distribution.
Parameters Probability density function Cumulative distribution function $\mu$ location; $c > 0\,$ scale $x \in [\mu, \infty)$ $\sqrt{\frac{c}{2\pi}}~~\frac{e^{-\frac{c}{2(x-\mu)}}}{(x-\mu)^{3/2}}$ $\textrm{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right)$ $\infty$ $c/2(\textrm{erfc}^{-1}(1/2))^2\,$, for $\mu=0$ $\frac{c}{3}$, for $\mu=0$ $\infty$ undefined undefined $\frac{1+3\gamma+\ln(16\pi c^2)}{2}$ where $\gamma$ is Euler's constant undefined $e^{i\mu t-\sqrt{-2ict}}$

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 one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the normal distribution and the Cauchy 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

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

where $\textrm{erfc}(z)$ is the complementary error function. 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 Levy 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 $\beta=1$:

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

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

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

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:

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

which diverges for $t>0$ 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:

$\lim_{x\rightarrow \infty}f(x;\mu,c) =\sqrt{\frac{c}{2\pi}}~\frac{1}{x^{3/2}}.$

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 scale.

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

$\left\{f(x) (-c-3 \mu +3 x)+2 (x-\mu )^2 f'(x)=0,f(0)=\frac{e^{\frac{c}{2 \mu }} \left(-\frac{c}{\mu }\right)^{3/2}}{\sqrt{2 \pi } c}\right\}$

Related distributions

• If $X \sim \textrm{Levy}(\mu,c)\,$ then $k X + b \sim \textrm{Levy}(k \mu + b ,k c) \,$
• If $X\,\sim\,\textrm{Levy}(0,c)$ then $X\,\sim\,\textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2})$ (inverse gamma distribution)
• Lévy distribution is a special case of type 5 Pearson distribution
• If $Y\,\sim\,\textrm{Normal}(\mu,\sigma^2)$ (Normal distribution) then ${(Y-\mu)}^{-2} \sim\,\textrm{Levy}(0,1/\sigma^2)$
• If $X \sim \textrm{Normal}(\mu,\tfrac{1}{\sqrt{\sigma}})\,$ then ${(X-\mu)}^{-2} \sim \textrm{Levy}(0,\sigma)\,$
• If $X\,\sim\,\textrm{Levy}(\mu,c)$ then $X\,\sim\,\textrm{Stable}(1/2,1,c,\mu) \,$ (Stable distribution)
• If $X\,\sim\,\textrm{Levy}(0,c)$ then $X\,\sim\,\textrm{Scale-inv-}\chi^2(1,c)$ (Scaled-inverse-chi-squared distribution)
• If $X\,\sim\,\textrm{Levy}(\mu,c)$ then ${(X-\mu)}^{-\tfrac{1}{2}} \sim\,\textrm{FoldedNormal}(0,1/\sqrt{c})$ (Folded normal distribution)

Applications

• The frequency of geomagnetic reversals appears to follow a Lévy distribution
• The time of hitting a single point $\alpha$ (different from the starting point 0) 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.[1]

Footnotes

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]

Notes

1. ^ Rogers, Geoffrey L, Multiple path analysis of reflectance from turbid media. Journal of the Optical Society of America A, 25:11, p 2879-2883 (2008).
2. ^ Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes". University of Sheffield. pp. 37–53.