Lévy distribution

For the more general family of Lévy alpha-stable distributions, of which this distribution is a special case, see stable 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.^{[note 1]} It is a special case of the inverse-gamma distribution. It is a stable distribution.

Lévy (unshifted)

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu }$ location; ${\displaystyle c>0\,}$ scale
Support	${\displaystyle x\in (\mu ,\infty )}$
PDF	${\displaystyle {\sqrt {\frac {c}{2\pi }}}~~{\frac {e^{-{\frac {c}{2(x-\mu )}}}}{(x-\mu )^{3/2}}}}$
CDF	${\displaystyle {\textrm {erfc}}\left({\sqrt {\frac {c}{2(x-\mu )}}}\right)}$
Quantile	${\displaystyle \mu +{\frac {\sigma }{2\left({\textrm {erfc}}^{-1}(p)\right)^{2}}}}$
Mean	${\displaystyle \infty }$
Median	${\displaystyle \mu +c/2({\textrm {erfc}}^{-1}(1/2))^{2}\,}$
Mode	${\displaystyle \mu +{\frac {c}{3}}}$
Variance	${\displaystyle \infty }$
Skewness	undefined
Excess 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	${\displaystyle e^{i\mu t-{\sqrt {-2ict}}}}$

Definition

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

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

where ${\displaystyle \operatorname {erfc} (z)}$ is the complementary error function, and ${\displaystyle \Phi (x)}$ 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 [${\displaystyle \mu }$, ${\displaystyle \infty }$). 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

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

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

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

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

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

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

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

however, this diverges for ${\displaystyle t>0}$ 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 ${\displaystyle x\to \infty ,}$

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:

Probability density function for the Lévy distribution on a log–log plot

The standard Lévy distribution satisfies the condition of being stable:

${\displaystyle (X_{1}+X_{2}+\dotsb +X_{n})\sim n^{1/\alpha }X,}$

where ${\displaystyle X_{1},X_{2},\ldots ,X_{n},X}$ are independent standard Lévy-variables with ${\displaystyle \alpha =1/2.}$

Related distributions

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

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

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

Footnotes

1. "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,