Lévy distribution

Lévy (unshifted)

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu }$ location; ${\displaystyle c>0}$ scale
Support	${\displaystyle x\in (\mu ,\mathrm{\infty })}$
PDF	${\displaystyle \sqrt{\frac{c}{2\pi }}\frac{e^{-\frac{c}{2(x-\mu )}}}{(x-\mu {)}^{3/2}}}$
CDF	${\displaystyle \text{<mrow data-mjx-texclass="ORD">erfc</mrow>}\left(\sqrt{\frac{c}{2(x-\mu )}}\right)}$
Quantile	${\displaystyle \mu +\frac{\sigma }{2{({\text{<mrow data-mjx-texclass="ORD">erfc</mrow>}}^{-1}(p))}^2}}$
Mean	${\displaystyle \mathrm{\infty }}$
Median	${\displaystyle \mu +c/2({\text{<mrow data-mjx-texclass="ORD">erfc</mrow>}}^{-1}(1/2){)}^2}$
Mode	${\displaystyle \mu +\frac{c}{3}}$
Variance	${\displaystyle \mathrm{\infty }}$
Skewness	undefined
Kurtosis	undefined
Entropy	${\displaystyle \frac{1+3\gamma +\ln (16\pi c^2)}{2}}$ where ${\displaystyle \gamma }$ is the Euler-Mascheroni constant
MGF	undefined
CF	${\displaystyle e^{i\mu t-\sqrt{-2ict}}}$

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 a stable distribution.

The probability density function of the Lévy distribution over the domain ${\displaystyle x\ge \mu }$ is

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

where ${\displaystyle \mu }$ is the location parameter, and ${\displaystyle c}$ is the scale parameter. The cumulative distribution function is

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

where ${\displaystyle \mathrm{erfc}(z)}$ is the complementary error function, and ${\displaystyle \mathrm{\Phi }(x)}$ is the Laplace function (CDF of the standard normal distribution). The shift parameter ${\displaystyle \mu }$ has the effect of shifting the curve to the right by an amount ${\displaystyle \mu }$ and changing the support to the interval [${\displaystyle \mu }$, ${\displaystyle \mathrm{\infty }}$). Like all stable distributions, the Lévy distribution has a standard form f(x; 0, 1) which has the following property:

${\displaystyle f(x;\mu ,c)dx=f(y;0,1)dy,}$

where y is defined as

${\displaystyle y=\frac{x-\mu }{c}.}$

The characteristic function of the Lévy distribution is given by

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

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

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

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

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

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

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

however, this diverges for ${\displaystyle t>0}$ 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:

${\displaystyle f(x;\mu ,c)\sim \sqrt{\frac{c}{2\pi }}\frac{1}{x^{3/2}}}$ as ${\displaystyle x\to \mathrm{\infty },}$

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 ${\displaystyle \mu =0}$ are plotted on a log–log plot:

The standard Lévy distribution satisfies the condition of being stable:

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

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

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

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]

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

is Lévy-distributed with location ${\displaystyle \mu }$ and scale ${\displaystyle c}$. Here ${\displaystyle \mathrm{\Phi }(x)}$ is the cumulative distribution function of the standard normal distribution.

• The frequency of geomagnetic reversals appears to follow a Lévy distribution
• The time of hitting a single point, at distance ${\displaystyle \alpha }$ from the starting point, by the Brownian motion has the Lévy distribution with ${\displaystyle c={\alpha }^2}$. (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]

• "Information on stable distributions". https://edspace.american.edu/jpnolan/stable/.  - 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

• Weisstein, Eric W.. "Lévy Distribution". http://mathworld.wolfram.com/LevyDistribution.html. 

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