, for these demonstrations so that scaling is consistent with the normal distribution. When α is near 1 computations run slowly.
Stable Tail Behavior
In the following demonstrations the 0-parameterization of the stable distribution is used to keep the plots centered near zero when α is near 1. Log log plots are used to show the full range of tail behavior which for α < 2 eventually becomes log log linear distally. The first plot below shows the right tail of the stable distribution in red and the normal distribution in blue for comparison. When α = 2, they will be superimposed. When α is just below 2, the stable curve will hug the normal centrally then sharply break away far out on the tail, the points α = 1.9, α = 1.99, and α = 1.999, have been bookmarked and can be accessed by clicking the icon in the upper right corner. Mathematica has some difficulty calculating the tail at α = 1.999. For lower values of α the stable tail diverges far from the normal early in the distribution. The normal distribution has σ = , for these demonstrations so that scaling is consistent with the normal distribution. When α is near 1 computations run slowly.
The next graphic looks at the right tail of the stable distribution is maximally left skewed with β = -1. When α = 2, the distribution is not skewed, and the tail matches the normal. Elsewhere with 1 ≤ α < 2, the tail becomes progressively lighter than the normal as α → 1.
The last picture shows both tails with the left tail reversed to be parallel to the right. A larger range of α is shown down to 0.2. The left and right tails are superimposed when β = 0, the tails are otherwise parallel until β = ± 1, when the light tail becomes lighter than the normal.
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© Copyright 2011 Robert H. Rimmer 24 July 2011
