# 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 $x\geq \mu$ 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:
    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.

## 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,