Levy skew alpha-stable distribution

A Lévy skew alpha-stable distribution or just stable distribution is a distribution where sums of random variables have the same distribution as the original. If $X 1, X 2$ are stable and independently and identically distributed (iid) and if $Y = a X 1 + b X 2 + c$ is a linear combination of the two, then Y is of the same type: $Y = d X + e$ . If e=0 for all a, b and c it is called strictly stable.

Levy distributions are found in analysis of critical behavior and financial data. It was developed by and named after the French mathematician Paul Lévy.

Contents

1 The distribution

2 Special cases

3 Stability property

4 The generalized central limit theorem

5 See also

6 References

The distribution

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 $\varphi(t)$ :

$p(\alpha,\beta,c,\mu;\,x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} \varphi (t) e^{-itx}\,dt$

where $\varphi(t)$ is

$\varphi(t) = \exp\left[~it\mu - |c t|^\alpha\,(1-i \beta\,\textrm{sign}(t) \tan(\pi \alpha/2))~\right] ~~~~~~0<\alpha\le 2, -1\le\beta\le 1$

When α = 1 the term

$\tan(\pi \alpha/2)\,$

is replaced by

$-(2/\pi)\log|t|.\,$

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

$\lim_{|x|\rightarrow\infty}p(x)=\frac{\alpha C^\alpha}{|x|^{1+\alpha}}$

where C is proportional to c. This "power law tail" behavior causes the variance of Lévy distributions to be infinite for all α< 2.

Special cases

There is no general explicit solution for the form of p(x). However, a number of special cases can be found by inspection of the characteristic function.

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.
In the limit as c approaches zero or as α approaches zero the distribution will approach a Dirac delta function δ(x-μ).

Stability property

The Lévy alpha-stable distributions have the "stability" property that if N alpha-stable variates X_i are drawn from the distribution

$p(\alpha, \beta,c,\mu;\,X)\,$

then the sum

$Y = \sum_{i=1}^N k_i (X_i-\mu)\,$

will also be distributed as an alpha-stable variate,

$p(Y)=\frac{1}{S}\,\,p(\alpha, \beta,c,0;\,Y/S).\,$

where

$S=\left(\sum_{i=1}^N |k_i|^\alpha\right)^{1/\alpha}.\,$

This can be easily proven using the properties of characteristic functions.

The generalized central limit theorem

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.

References

GNU Scientific Library - Reference Manual Edition 1.6, for GSL Version 1.6, 27 December 2004
- "The Levy skew alpha-Stable Distribution." GNU Scientific Library - Reference Manual. Accessed on February 11, 2005.
See http://www.e.u-tokyo.ac.jp/cirje/research/dp/2004/2004cf292.pdf for an improved numerical estimation of the Lévy symmetric stable distribution function.

Categories: Probability distributions