Levy function

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Lévy (unshifted)
Probability density function
Levy distribution PDF
Cumulative distribution function
Levy distribution CDF
Parameters μ {\displaystyle \mu } {\displaystyle \mu } location; c > 0 {\displaystyle c>0\,} {\displaystyle c>0\,} scale
Support x ∈ ( μ , ∞ ) {\displaystyle x\in (\mu ,\infty )} {\displaystyle x\in (\mu ,\infty )}
PDF c 2 π     e − c 2 ( x − μ ) ( x − μ ) 3 / 2 {\displaystyle {\sqrt {\frac {c}{2\pi }}}~~{\frac {e^{-{\frac {c}{2(x-\mu )}}}}{(x-\mu )^{3/2}}}} {\displaystyle {\sqrt {\frac {c}{2\pi }}}~~{\frac {e^{-{\frac {c}{2(x-\mu )}}}}{(x-\mu )^{3/2}}}}
CDF erfc ( c 2 ( x − μ ) ) {\displaystyle {\textrm {erfc}}\left({\sqrt {\frac {c}{2(x-\mu )}}}\right)} {\displaystyle {\textrm {erfc}}\left({\sqrt {\frac {c}{2(x-\mu )}}}\right)}
Quantile μ + σ 2 ( erfc − 1 ( p ) ) 2 {\displaystyle \mu +{\frac {\sigma }{2\left({\textrm {erfc}}^{-1}(p)\right)^{2}}}} {\displaystyle \mu +{\frac {\sigma }{2\left({\textrm {erfc}}^{-1}(p)\right)^{2}}}}
Mean ∞ {\displaystyle \infty } {\displaystyle \infty }
Median μ + c / 2 ( erfc − 1 ( 1 / 2 ) ) 2 {\displaystyle \mu +c/2({\textrm {erfc}}^{-1}(1/2))^{2}\,} {\displaystyle \mu +c/2({\textrm {erfc}}^{-1}(1/2))^{2}\,}
Mode μ + c 3 {\displaystyle \mu +{\frac {c}{3}}} {\displaystyle \mu +{\frac {c}{3}}}
Variance ∞ {\displaystyle \infty } {\displaystyle \infty }
Skewness undefined
Excess kurtosis undefined
Entropy

1 + 3 γ + ln ⁡ ( 16 π c 2 ) 2 {\displaystyle {\frac {1+3\gamma +\ln(16\pi c^{2})}{2}}} {\displaystyle {\frac {1+3\gamma +\ln(16\pi c^{2})}{2}}}

where γ {\displaystyle \gamma } {\displaystyle \gamma } is the Euler-Mascheroni constant
MGF undefined
CF e i μ t − − 2 i c t {\displaystyle e^{i\mu t-{\sqrt {-2ict}}}} {\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 and a stable distribution.

Definition

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

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

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

F ( x ; μ , c ) = erfc ⁡ ( c 2 ( x − μ ) ) = 2 − 2 Φ ( c ( x − μ ) ) , {\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 erfc ⁡ ( z ) {\displaystyle \operatorname {erfc} (z)} {\displaystyle \operatorname {erfc} (z)} is the complementary error function, and Φ ( x ) {\displaystyle \Phi (x)} {\displaystyle \Phi (x)} is the Laplace function (CDF of the standard normal distribution). The shift parameter μ {\displaystyle \mu } {\displaystyle \mu } has the effect of shifting the curve to the right by an amount μ {\displaystyle \mu } {\displaystyle \mu } and changing the support to the interval [ μ {\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:

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

where y is defined as

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

The characteristic function of the Lévy distribution is given by

φ ( t ; μ , c ) = e i μ t − − 2 i c t . {\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 α = 1 / 2 {\displaystyle \alpha =1/2} {\displaystyle \alpha =1/2} and β = 1 {\displaystyle \beta =1} {\displaystyle \beta =1}:

φ ( t ; μ , c ) = e i μ t − | c t | 1 / 2 ( 1 − i sign ⁡ ( t ) ) . {\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 μ = 0 {\displaystyle \mu =0} {\displaystyle \mu =0}, the nth moment of the unshifted Lévy distribution is formally defined by

m n   = def   c 2 π ∫ 0 ∞ e − c / 2 x x n x 3 / 2 d x , {\displaystyle m_{n}\ {\stackrel {\text{def}}{=}}\ {\sqrt {\frac {c}{2\pi }}}\int _{0}^{\infty }{\frac {e^{-c/2x}x^{n}}{x^{3/2}}}\,dx,} {\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 n ≥ 1 / 2 {\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

M ( t ; c )   = d e f   c 2 π ∫ 0 ∞ e − c / 2 x + t x x 3 / 2 d x , {\displaystyle M(t;c)\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {c}{2\pi }}}\int _{0}^{\infty }{\frac {e^{-c/2x+tx}}{x^{3/2}}}\,dx,} {\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 t > 0 {\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:

f ( x ; μ , c ) ∼ c 2 π 1 x 3 / 2 {\displaystyle f(x;\mu ,c)\sim {\sqrt {\frac {c}{2\pi }}}\,{\frac {1}{x^{3/2}}}} {\displaystyle f(x;\mu ,c)\sim {\sqrt {\frac {c}{2\pi }}}\,{\frac {1}{x^{3/2}}}} as x → ∞ , {\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 μ = 0 {\displaystyle \mu =0} {\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:

( X 1 + X 2 + ⋯ + X n ) ∼ n 1 / α X , {\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 X 1 , X 2 , … , X n , X {\displaystyle X_{1},X_{2},\ldots ,X_{n},X} {\displaystyle X_{1},X_{2},\ldots ,X_{n},X} are independent standard Lévy-variables with α = 1 / 2. {\displaystyle \alpha =1/2.} {\displaystyle \alpha =1/2.}

  • If X ∼ Levy ⁡ ( μ , c ) {\displaystyle X\sim \operatorname {Levy} (\mu ,c)} {\displaystyle X\sim \operatorname {Levy} (\mu ,c)}, then k X + b ∼ Levy ⁡ ( k μ + b , k c ) . {\displaystyle kX+b\sim \operatorname {Levy} (k\mu +b,kc).} {\displaystyle kX+b\sim \operatorname {Levy} (k\mu +b,kc).}
  • If X ∼ Levy ⁡ ( 0 , c ) {\displaystyle X\sim \operatorname {Levy} (0,c)} {\displaystyle X\sim \operatorname {Levy} (0,c)}, then X ∼ I n v - G a m m a ⁡ ( 1 / 2 , c / 2 ) {\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 Y ∼ Normal ⁡ ( μ , σ 2 ) {\displaystyle Y\sim \operatorname {Normal} (\mu ,\sigma ^{2})} {\displaystyle Y\sim \operatorname {Normal} (\mu ,\sigma ^{2})} (normal distribution), then ( Y − μ ) − 2 ∼ Levy ⁡ ( 0 , 1 / σ 2 ) . {\displaystyle (Y-\mu )^{-2}\sim \operatorname {Levy} (0,1/\sigma ^{2}).} {\displaystyle (Y-\mu )^{-2}\sim \operatorname {Levy} (0,1/\sigma ^{2}).}
  • If Y ∼ Normal ⁡ ( μ , 1 / c ) {\displaystyle Y\sim \operatorname {Normal} (\mu ,1/c)} {\displaystyle Y\sim \operatorname {Normal} (\mu ,1/c)}, then ( Y − μ ) − 2 ∼ Levy ⁡ ( 0 , c ) {\displaystyle (Y-\mu )^{-2}\sim \operatorname {Levy} (0,c)} {\displaystyle (Y-\mu )^{-2}\sim \operatorname {Levy} (0,c)}.
  • If X ∼ Levy ⁡ ( μ , c ) {\displaystyle X\sim \operatorname {Levy} (\mu ,c)} {\displaystyle X\sim \operatorname {Levy} (\mu ,c)}, then X ∼ Stable ⁡ ( 1 / 2 , 1 , c , μ ) {\displaystyle X\sim \operatorname {Stable} (1/2,1,c,\mu )} {\displaystyle X\sim \operatorname {Stable} (1/2,1,c,\mu )} (stable distribution).
  • If X ∼ Levy ⁡ ( 0 , c ) {\displaystyle X\sim \operatorname {Levy} (0,c)} {\displaystyle X\sim \operatorname {Levy} (0,c)}, then X ∼ S c a l e - i n v - χ 2 ⁡ ( 1 , c ) {\displaystyle X\,\sim \,\operatorname {Scale-inv-\chi ^{2}} (1,c)} {\displaystyle X\,\sim \,\operatorname {Scale-inv-\chi ^{2}} (1,c)} (scaled-inverse-chi-squared distribution).
  • If X ∼ Levy ⁡ ( μ , c ) {\displaystyle X\sim \operatorname {Levy} (\mu ,c)} {\displaystyle X\sim \operatorname {Levy} (\mu ,c)}, then ( X − μ ) − 1 / 2 ∼ FoldedNormal ⁡ ( 0 , 1 / c ) {\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-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]

X = F − 1 ( U ) = c ( Φ − 1 ( 1 − U / 2 ) ) 2 + μ {\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 μ {\displaystyle \mu } {\displaystyle \mu } and scale c {\displaystyle c} {\displaystyle c}. Here Φ ( x ) {\displaystyle \Phi (x)} {\displaystyle \Phi (x)} is the cumulative distribution function of the standard normal distribution.

Applications

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,

Notes

  1. "The Lévy Distribution". Random. Probability, Mathematical Statistics, Stochastic Processes. The University of Alabama in Huntsville, Department of Mathematical Sciences. Archived from the original on 2017-08-02.
  2. Rogers, Geoffrey L. (2008). "Multiple path analysis of reflectance from turbid media". Journal of the Optical Society of America A. 25 (11): 2879–2883. Bibcode:2008JOSAA..25.2879R. doi:10.1364/josaa.25.002879. PMID 18978870.
  3. Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.

References