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.
More intriguing information
1. The name is absent2. The name is absent
3. On Dictatorship, Economic Development and Stability
4. Density Estimation and Combination under Model Ambiguity
5. Qualification-Mismatch and Long-Term Unemployment in a Growth-Matching Model
6. The name is absent
7. Better policy analysis with better data. Constructing a Social Accounting Matrix from the European System of National Accounts.
8. L'organisation en réseau comme forme « indéterminée »
9. The name is absent
10. The name is absent