Lévy 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.

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

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

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

${\displaystyle F(x;\mu ,c)=\operatorname {erfc} \left({\sqrt {\frac {c}{2(x-\mu )}}}\right)=2-2\Phi \left({\sqrt {\frac {c}{(x-\mu )}}}\right),}${\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 ${\displaystyle \operatorname {erfc} (z)}${\displaystyle \operatorname {erfc} (z)} is the complementary error function, and ${\displaystyle \Phi (x)}${\displaystyle \Phi (x)} is the Laplace function (CDF of the standard normal distribution). The shift parameter ${\displaystyle \mu }${\displaystyle \mu } has the effect of shifting the curve to the right by an amount ${\displaystyle \mu }${\displaystyle \mu } and changing the support to the interval [${\displaystyle \mu }${\displaystyle \mu }, ${\displaystyle \infty }${\displaystyle \infty }). Like all stable distributions, the Lévy distribution has a standard form f(x; 0, 1) which has the following property:

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

where y is defined as

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

The characteristic function of the Lévy distribution is given by

${\displaystyle \varphi (t;\mu ,c)=e^{i\mu t-{\sqrt {-2ict}}}.}${\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 ${\displaystyle \alpha =1/2}${\displaystyle \alpha =1/2} and ${\displaystyle \beta =1}${\displaystyle \beta =1}:

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

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

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

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

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

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

${\displaystyle f(x;\mu ,c)\sim {\sqrt {\frac {c}{2\pi }}},円{\frac {1}{x^{3/2}}}}${\displaystyle f(x;\mu ,c)\sim {\sqrt {\frac {c}{2\pi }}},円{\frac {1}{x^{3/2}}}} as ${\displaystyle x\to \infty ,}${\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 ${\displaystyle \mu =0}${\displaystyle \mu =0} are plotted on a log–log plot:

The standard Lévy distribution satisfies the condition of being stable:

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

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

• If ${\displaystyle X\sim \operatorname {Levy} (\mu ,c)}${\displaystyle X\sim \operatorname {Levy} (\mu ,c)}, then ${\displaystyle kX+b\sim \operatorname {Levy} (k\mu +b,kc).}${\displaystyle kX+b\sim \operatorname {Levy} (k\mu +b,kc).}
• If ${\displaystyle X\sim \operatorname {Levy} (0,c)}${\displaystyle X\sim \operatorname {Levy} (0,c)}, then ${\displaystyle X\sim \operatorname {Inv-Gamma} (1/2,c/2)}${\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 ${\displaystyle Y\sim \operatorname {Normal} (\mu ,\sigma ^{2})}${\displaystyle Y\sim \operatorname {Normal} (\mu ,\sigma ^{2})} (normal distribution), then ${\displaystyle (Y-\mu )^{-2}\sim \operatorname {Levy} (0,1/\sigma ^{2}).}${\displaystyle (Y-\mu )^{-2}\sim \operatorname {Levy} (0,1/\sigma ^{2}).}
• If ${\displaystyle X\sim \operatorname {Normal} (\mu ,1/{\sqrt {\sigma }})}${\displaystyle X\sim \operatorname {Normal} (\mu ,1/{\sqrt {\sigma }})}, then ${\displaystyle (X-\mu )^{-2}\sim \operatorname {Levy} (0,\sigma )}${\displaystyle (X-\mu )^{-2}\sim \operatorname {Levy} (0,\sigma )}.
• If ${\displaystyle X\sim \operatorname {Levy} (\mu ,c)}${\displaystyle X\sim \operatorname {Levy} (\mu ,c)}, then ${\displaystyle X\sim \operatorname {Stable} (1/2,1,c,\mu )}${\displaystyle X\sim \operatorname {Stable} (1/2,1,c,\mu )} (stable distribution).
• If ${\displaystyle X\sim \operatorname {Levy} (0,c)}${\displaystyle X\sim \operatorname {Levy} (0,c)}, then ${\displaystyle X,円\sim ,円\operatorname {Scale-inv-\chi ^{2}} (1,c)}${\displaystyle X,円\sim ,円\operatorname {Scale-inv-\chi ^{2}} (1,c)} (scaled-inverse-chi-squared distribution).
• If ${\displaystyle X\sim \operatorname {Levy} (\mu ,c)}${\displaystyle X\sim \operatorname {Levy} (\mu ,c)}, then ${\displaystyle (X-\mu )^{-1/2}\sim \operatorname {FoldedNormal} (0,1/{\sqrt {c}})}${\displaystyle (X-\mu )^{-1/2}\sim \operatorname {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]

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

is Lévy-distributed with location ${\displaystyle \mu }${\displaystyle \mu } and scale ${\displaystyle c}${\displaystyle c}. Here ${\displaystyle \Phi (x)}${\displaystyle \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 ${\displaystyle \alpha }${\displaystyle \alpha } from the starting point, by the Brownian motion has the Lévy distribution with ${\displaystyle c=\alpha ^{2}}${\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" . 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

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

• Continuous distributions
• Probability distributions with non-finite variance
• Power laws
• Stable distributions
• Paul Lévy (mathematician)

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