Properties
- All stable distributions are infinitely divisible.
- With the exception of the normal distribution (α = 2), stable distributions are leptokurtotic and heavy-tailed distributions.
- Closure under convolution
Stable distributions are closed under convolution for a fixed value of α. Since convolution is equivalent to multiplication of the Fourier-transformed function, it follows that the product of two stable characteristic functions with the same α will yield another such characteristic function. The product of two stable characteristic functions is given by:
Since Φ is not a function of the μ, c or β variables it follows that these parameters for the convolved function are given by:
In each case, it can be shown that the resulting parameters lie within the required intervals for a stable distribution.
Read more about this topic: Stable Distribution
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