Lévy distribution

Template:Short description Template:Probability 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

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

where $\mathrm{erfc}(z)$ is the complementary error function, and $\mathrm{\Phi }(x)$ 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 [$\mu$, $\mathrm{\infty }$). Like all stable distributions, the Lévy distribution has a standard form Template:Nobr 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

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

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

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

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

$m_n\stackrel{\text{def}}{=}\sqrt{\frac{c}{2\pi }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-c/2x}{x}^n}{x^{3/2}}dx,$

which diverges for all $n\ge 1/2$, 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

$M(t;c)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\frac{c}{2\pi }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-c/2x+tx}}{x^{3/2}}dx,$

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 $x\to \mathrm{\infty },$

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:

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:

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

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

Related distributions

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

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

is Lévy-distributed with location $\mu$ and scale $c$. Here $\mathrm{\Phi }(x)$ 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.

References

• - 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