1 Stable distributions
Moreover, the tails of α-stable distribution functions exhibit a crossover from
an approximate power decay with exponent α > 2 to the true tail with exponent
α. This phenomenon is more visible for large α’s (Weron, 2001).
When α > 1, the mean of the distribution exists and is equal to μ. In general,
the pth moment of a stable random variable is finite if and only if p < α. When
the skewness parameter β is positive, the distribution is skewed to the right,
i.e. the right tail is thicker, see the left panel of Figure 1.2. When it is negative,
it is skewed to the left. When β = 0, the distribution is symmetric about μ. As
α approaches 2, β loses its effect and the distribution approaches the Gaussian
distribution regardless of β. The last two parameters, σ and μ, are the usual
scale and location parameters, i.e. σ determines the width and μ the shift of
the mode (the peak) of the density. For σ = 1 and μ = 0 the distribution is
called standard stable.
1.3.1 Characteristic function representation
Due to the lack of closed form formulas for densities for all but three dis-
tributions (see the right panel in Figure 1.2), the α-stable law can be most
conveniently described by its characteristic function φ(t ) - the inverse Fourier
transform of the probability density function. However, there are multiple pa-
rameterizations for α-stable laws and much confusion has been caused by these
different representations, see Figure 1.3. The variety of formulas is caused by
a combination of historical evolution and the numerous problems that have
been analyzed using specialized forms of the stable distributions. The most
popular parameterization of the characteristic function of X ~ Sα( σ,β,μ ),
i.e. an α-stable random variable with parameters α, σ, β, and μ, is given by
(Samorodnitsky and Taqqu, 1994; Weron, 2004):
{—σα∣t∣α{ 1 — iβsign(t) tan πα} + iμt, α = 1,
(1.2)
—σ∣t∣{ 1 + iβsign(t)∏ ln ∣t∣} + iμt, α = 1.
For numerical purposes, it is often advisable to use Nolan’s (1997) parameter-
ization:
{—σα∣t∣α{ 1 + iβ sign( t ) tan πβ- [( σ∣t∣ )1 α — 1]} + iμ 01, α=1,
—σ∣t∣{ 1 + iβsign(t)∏ ln(σ∣t∣)} + iμ01, α = 1.
(1.3)