Stable Distribution

Synonyms

Lévy skew α-stable distribution

Definition

A random variable Z is said to follow a symmetric α-stable distribution [13, 15], where 0 < α ≤ 2, if the Fourier transform of its probability density function f _Z (z) satisfies

$$ {\int}_{-\infty}^{\infty }{e}^{\sqrt{-1} zt}{f}_Z(z) dt={e}^{-d\left|t\right|{}^{\alpha }}, 0<\alpha \le 2 $$

(1)

where d > 0 is the scale parameter. This is denoted by Z ∼ S(α, d).

There is an equivalent definition. A random variable Z follows a symmetric α-stable distribution if, for any real numbers, C ₁ and C ₂,

$$ {C}_1{Z}_1+{C}_2{Z}_2\overset{d}{=}{\left(|{C}_1|{}^{\alpha }+|{C_2}^{\alpha}\right)}^{1/\alpha }Z, $$

(2)

where Z ₁ and Z ₂ are independent copies of Z, and the symbol “$\overset{d}{=}$” denotes equality in distribution.

The probability density function f _Z (z) can be obtained by taking inverse Fourier transform of 1. In particular, f _Z (z) can be expressed in closed-forms when α = 2 (i.e., the normal distribution) and α= 1 (i.e., the...