Stable Distributions in Statistical Inference: 1. Symmetric Stable Distributions Compared to Other Symmetric Long-Tailed Distributions

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Abstract
The hypothesis that very long-tailed data come from a symmetric stable distribution can be generalized, namely, that the observations are generated by a mixture of a normal distribution and another symmetric stable distribution. The difficulty of statistical distinction of the two cases is computed as a function of the various parameters involved. Also, computations show that convolutions of these mixtures can be very slow to converge to the limiting stable distribution or even to assume the asymptotic behavior predicted by theory for large numbers of convolutions.

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