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

Lévy (unshifted)

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

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.

References

• "Information on stable distributions". Retrieved September 5, 2021. - 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

External links

• Weisstein, Eric W. "Lévy Distribution". MathWorld.