The Distribution
A stable distribution is therefore specified by the above four parameters. It can be shown that any non-degenerate stable distribution has a smooth (infinitely differentiable) density function.(Nolan 2009, Theorem 1.9) If denotes the density of X and Y is the sum of independent copies of X:
then Y has the density with
The asymptotic behavior is described, for α< 2, by: (Nolan 2009, Theorem 1.12)
where Γ is the Gamma function (except that when α < 1 and β = ±1, the tail vanishes to the left or right, resp., of μ). This "heavy tail" behavior causes the variance of stable distributions to be infinite for all α < 2. This property is illustrated in the log-log plots below.
When α = 2, the distribution is Gaussian (see below), with tails asymptotic to exp(−x2/4c2)/(2c√π).
Read more about this topic: Stable Distribution
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