A set of four symmetric centered Lévy distributions with scale factor c=1. Red: α=0.5, Blue: α=1.0, Green: α=1.5, Black: α=2.0. A Lévy skew alpha-stable distribution is a probability distribution function which
is found in analysis of critical behavior and financial data. It was developed by
and named after the French mathematician Paul Lévy. A Lévy skew stable distribution is specified by scale c, exponent α, shift μ and skewness parameter β. The skewness parameter must lie in the range [-1,1] and when it is zero, the distribution is symmetric and is referred to as a Lévy symmetric alpha-stable distribution. The exponent α must lie in the range (0,2].
The Lévy skew stable probability distribution is defined by the Fourier transform of its characteristic function <math>\varphi(t)<math>:
- <math>
p(\alpha,\beta,c,\mu;\,x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} \varphi (t)
e^{-itx}dt
<math>
where <math>\varphi(t)<math> is
- <math>
\varphi(t) =
\exp\left[~it\mu - |c t|^\alpha\,(1-i \beta\,\textrm{sign}(t) \tan(\pi \alpha/2))~\right]
~~~~~~0<\alpha\le 2, -1\le\beta\le 1
<math>
When α = 1 the term
- <math>\tan(\pi \alpha/2)\,<math>
is replaced by
- <math>-(2/\pi)\log|t|\,<math>.
μ is the location of the peak of the distribution.
β is a measure of asymmetry, with β=0 yielding a distribution symmetric
about x=μ.
c is a scale factor which is a measure of the width of the distribution and
α is the exponent or index of the distribution and specifies the asymptotic behavior of
the distribution for α<2
- <math>
\lim_{|x|\rightarrow\infty}p(x)=\frac{\alpha C^\alpha}{|x|^{1+\alpha}}
<math>
where C is proportional to c. This "power law tail" behavior
causes the variance of Lévy distributions to be infinite for all α< 2.
There is no general explicit solution for the form of p(x).
However, for α=2 the distribution reduces to a Gaussian distribution with
variance σ2 = 2c2
and mean μ and the skewness parameter β has no effect.
For α=1 and β=0 the distribution
reduces to a Cauchy distribution with scale parameter c and shift parameter μ.
The Lévy alpha-stable distributions have the "stability" property that if N alpha-stable variates Xi are drawn from the distribution
- <math>p(\alpha, \beta,c,\mu;\,X)\,<math>
then the sum
- <math>Y = \sum_{i=1}^N k_i (X_i-\mu)\,<math>
will also be distributed as an alpha-stable variate,
- <math>p(Y)=\frac{1}{S}\,\,p(\alpha, \beta,c,0;\,Y/S)\,<math>.
where
- <math>S=\left(\sum_{i=1}^N |k_i|^\alpha\right)^{1/\alpha}<math>
This can be easily proven using the properties of characteristic functions.
Another important property of Lévy distributions is the role that they play in
a generalized central limit theorem. The central limit theorem states that
the sum of a number of random variables with finite variances will tend to a
normal distribution as the number of variables grows. A generalization
due to Gnedenko and Kolmogorov states that
the sum of a number of random
variables with power-law tail distributions decreasing as 1/|x|α+1 (and
therefore having infinite variance) will tend to a stable Levy distribution p(α,0,c,0; x) as the number of variables grows.
See Also
References
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