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. SOME ISSUES IN LAND TENURE, OWNERSHIP AND CONTROL IN DISPERSED VS. CONCENTRATED AGRICULTURE
3. El Mercosur y la integración económica global
4. Volunteering and the Strategic Value of Ignorance
5. A parametric approach to the estimation of cointegration vectors in panel data
6. Biological Control of Giant Reed (Arundo donax): Economic Aspects
7. THE WAEA -- WHICH NICHE IN THE PROFESSION?
8. SLA RESEARCH ON SELF-DIRECTION: THEORETICAL AND PRACTICAL ISSUES
9. Modelling Transport in an Interregional General Equilibrium Model with Externalities
10. Developmental Robots - A New Paradigm