Lévy 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. It is a special case of the inverse-gamma distribution and a stable distribution.

Definition

The probability density function of the Lévy distribution over the domain <math>x \ge \mu</math> is

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

<math display=block>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 which has the following property:

where y is defined as

The characteristic function of the Lévy distribution is given by

<math display=block>\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 display=block>\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 display=block>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 display=block>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 display=block>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:

325px|thumb|center|Probability density function for the Lévy distribution on a log–log plot

The standard Lévy distribution satisfies the condition of being stable:

<math display=block>(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>

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>Y \sim \operatorname{Normal}(\mu, 1/c)</math>, then <math>(Y - \mu)^{-2} \sim \operatorname{Levy}(0, c)</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

: <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.
A Cauchy process can be defined as a Brownian motion subordinated to a process associated with a Lévy distribution.

Footnotes

Notes

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