Lévy distribution

Template:Short description Template:For Template:Probability distribution~~\frac{e^{-\frac{c}{2(x-\mu)}}}{(x-\mu)^{3/2}}</math>|

 cdf        =<math>\textrm{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right)</math>|
 quantile   =<math>\mu+\frac{\sigma}{2\left(\textrm{erfc}^{-1}(p)\right)^2}</math>|
 mean       =<math>\infty</math>|
 median     =<math>\mu+c/2(\textrm{erfc}^{-1}(1/2))^2\,</math>|
 mode       =<math>\mu + \frac{c}{3}</math>|
 variance   =<math>\infty</math>|
 skewness   =undefined|
 kurtosis   =undefined|
 entropy    =<math>\frac{1+3\gamma+\ln(16\pi c^2)}{2}</math>

where <math>\gamma</math> is the Euler-Mascheroni constant|

 mgf        =undefined|
 char       =<math>e^{i\mu t-\sqrt{-2ict}}</math>|

}} 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.<ref group="note">"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, Template:Isbn, Template:Isbn, [1]; and in Journal of technical physics, Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995, [2]</ref> 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 <math>x \ge \mu</math> is

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

where <math>\mu</math> is the location parameter, and <math>c</math> is the scale parameter. The cumulative distribution function is

<math>F(x; \mu, c) = \operatorname{erfc}\left(\sqrt{\frac{c}{2(x - \mu)}}\right) = 2 - 2 \Phi\left({\sqrt{\frac{c}{(x - \mu)}}}\right),</math>

where <math>\operatorname{erfc}(z)</math> is the complementary error function, and <math>\Phi(x)</math> is the Laplace function (CDF of the standard normal distribution). The shift parameter <math>\mu</math> has the effect of shifting the curve to the right by an amount <math>\mu</math> and changing the support to the interval [<math>\mu</math>, <math>\infty</math>). Like all stable distributions, the Lévy distribution has a standard form Template:Nobr which has the following property:

<math>f(x; \mu, c) \,dx = f(y; 0, 1) \,dy,</math>

where y is defined as

<math>y = \frac{x - \mu}{c}.</math>

The characteristic function of the Lévy distribution is given by

<math>\varphi(t; \mu, c) = e^{i\mu t - \sqrt{-2ict}}.</math>

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

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

Assuming <math>\mu = 0</math>, the nth moment of the unshifted Lévy distribution is formally defined by

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

which diverges for all <math>n \geq 1/2</math>, 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

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

however, this diverges for <math>t > 0</math> 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:

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

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 <math>\mu = 0</math> are plotted on a log–log plot:

File:Levy0 LdistributionPDF.svg

Probability density function for the Lévy distribution on a log–log plot

The standard Lévy distribution satisfies the condition of being stable:

<math>(X_1 + X_2 + \dotsb + X_n) \sim n^{1/\alpha}X,</math>

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

Related distributions

• If <math>X \sim \operatorname{Levy}(\mu, c)</math>, then <math>kX + b \sim \operatorname{Levy}(k\mu + b, kc).</math>
• If <math>X \sim \operatorname{Levy}(0, c)</math>, then <math>X \sim \operatorname{Inv-Gamma}(1/2, c/2)</math> (inverse gamma distribution). Here, the Lévy distribution is a special case of a Pearson type V distribution.
• If <math>Y \sim \operatorname{Normal}(\mu, \sigma^2)</math> (normal distribution), then <math>(Y - \mu)^{-2} \sim \operatorname{Levy}(0, 1/\sigma^2).</math>
• If <math>X \sim \operatorname{Normal}(\mu, 1/\sqrt{\sigma})</math>, then <math>(X - \mu)^{-2} \sim \operatorname{Levy}(0, \sigma)</math>.
• If <math>X \sim \operatorname{Levy}(\mu, c)</math>, then <math>X \sim \operatorname{Stable}(1/2, 1, c, \mu)</math> (stable distribution).
• If <math>X \sim \operatorname{Levy}(0, c)</math>, then <math>X\,\sim\,\operatorname{Scale-inv-\chi^2}(1, c)</math> (scaled-inverse-chi-squared distribution).
• If <math>X \sim \operatorname{Levy}(\mu, c)</math>, then <math>(X - \mu)^{-1/2} \sim \operatorname{FoldedNormal}(0, 1/\sqrt{c})</math> (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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>X = F^{-1}(U) = \frac{c}{(\Phi^{-1}(1 - U/2))^2} + \mu</math>

is Lévy-distributed with location <math>\mu</math> and scale <math>c</math>. Here <math>\Phi(x)</math> 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 <math>\alpha</math> from the starting point, by the Brownian motion has the Lévy distribution with <math>c=\alpha^2</math>. (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.<ref>Template:Cite journal</ref>
• A Cauchy process can be defined as a Brownian motion subordinated to a process associated with a Lévy distribution.<ref name=applebaum>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

References

• {{#invoke:citation/CS1|citation

|CitationClass=web }} - 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

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:LevyDistribution%7CLevyDistribution.html}} |title = Lévy Distribution |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}