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Additional information
By the way, this theory can also be applied, in the inverse form, to the case α<1, such as Zipf's linguistic distribution, which does not have finite mean and variance, (see e.g. Taguchi, T., On Zipf's law, Proc. Inst. Statist. Math., Vol. 17, No. 2, 1969).
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Taguchi, T. On Fechner's thesis and statistics with normp . Ann Inst Stat Math 26, 175–193 (1974). https://doi.org/10.1007/BF02479814
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DOI: https://doi.org/10.1007/BF02479814