1.3 Definitions and basic characteristics
Dependence on alpha
Figure 1.1: Left panel : A semilog plot of symmetric (β = μ = 0) α-stable
probability density functions (pdfs) for α = 2 (black solid line), 1.8
(red dotted line), 1.5 (blue dashed line) and 1 (green long-dashed
line). The Gaussian (α = 2) density forms a parabola and is the
only α-stable density with exponential tails. Right panel: Right
tails of symmetric α-stable cumulative distribution functions (cdfs)
for α = 2 (black solid line), 1.95 (red dotted line), 1.8 (blue dashed
line) and 1.5 (green long-dashed line) on a double logarithmic paper.
For α < 2 the tails form straight lines with slope -α.
ɑ STFstab01.xpl
exhibit a power-law behavior. More precisely, using a central limit theorem
type argument it can be shown that (Janicki and Weron, 1994; Samorodnitsky
and Taqqu, 1994):
limx→∞ xαP(X > x) = Cα(1 + β)σα,
(1.1)
limx→∞ xαP(X < -x) = Cα(1 + β)σα,
where:
Cα
=20
sin(x)dx
= 1Γ(α) sin πα.
π2
The convergence to a power-law tail varies for different α’s and, as can be seen
in the right panel of Figure 1.1, is slower for larger values of the tail index.
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