Relation between power laws and stable distributions.

Who can explain me the relation between a power law and the stable distributions? Namely How i can represent a power law in terms of an alpha-stable distribution?

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Well, for $\alpha<2$ stable distributions have tails behaving asymptotically as power laws. See Thm 1.12 here http://academic2.american.edu/~jpnolan/stable/chap1.pdf. As this is only an asymptotic property, I don't think you can represent a power law with a stable distribution.