Lévy distribution

Definition

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

${\displaystyle m_n\stackrel{\text{def}}{=}\sqrt{\frac{c}{2\pi }}{\int }_0^{\mathrm{\infty }}\frac{e^{-c/2x}{x}^n}{x^{3/2}}dx,}$

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

${\displaystyle M(t;c)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\frac{c}{2\pi }}{\int }_0^{\mathrm{\infty }}\frac{e^{-c/2x+tx}}{x^{3/2}}dx,}$

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

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 ${\displaystyle \alpha =1/2.}$