Lévy distribution

For the more general family of Lévy alpha-stable distributions, of which this distribution is a special case, see stable distribution.

Lévy (unshifted)
Probability density function
Cumulative distribution function
Parameters	$\mu$ location; $c > 0\,$ scale
Support	$x \in [\mu, \infty)$
PDF	$\sqrt{\frac{c}{2\pi}}~~\frac{e^{-\frac{c}{2(x-\mu)}}}{(x-\mu)^{3/2}}$
CDF	$\textrm{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right)$
Mean	$\infty$
Median	$c/2(\textrm{erfc}^{-1}(1/2))^2\,$ , for $\mu =0$
Mode	$\frac{c}{3}$ , for $\mu =0$
Variance	$\infty$
Skewness	undefined
Ex. kurtosis	undefined
Entropy	$\frac{1+3\gamma+\ln(16\pi c^2)}{2}$ where $\gamma$ is Euler's constant
MGF	undefined
CF	$e^{i\mu t-\sqrt{-2ict}}$

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:

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

x\to \infty .

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.

Differential equation

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

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)

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^[1]

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 $\alpha$ (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.^[2]
A Cauchy process can be defined as a Brownian motion subordinated to a process associated with a Lévy distribution.^[3]

Footnotes

↑ "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, ; and in Journal of technical physics, Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995,

Notes

↑ How to derive the function for a random sample from a Lévy Distribution: http://www.math.uah.edu/stat/special/Levy.html
↑ Rogers, Geoffrey L. (2008). "Multiple path analysis of reflectance from turbid media". Journal of the Optical Society of America A. 25 (11): 2879–2883.
↑ Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.

References

"Information on stable distributions". Retrieved July 13, 2005. - 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

Weisstein, Eric W. "Lévy Distribution". MathWorld.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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