Geodesic
Nonprobabilistic infinitely divisible distributions: the L\'evy--Khinchin representation, limit theorems
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 145-177

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We study properties of generalized infinitely divisible distributions with the Lévy measure $\Lambda(dx)=\frac{g(x)}{x^{1+\alpha}}\,dx$, $\alpha\in(2,4)\cup(4,6)$. Such measures are signed ones and hence they are not probability measures. We show that in some sence these signed measures are limit measures for sums of independent random variables. 
@article{ZNSL_2014_431_a9,
     author = {M. V. Platonova},
     title = {Nonprobabilistic infinitely divisible distributions: the {L\'evy--Khinchin} representation, limit theorems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {145--177},
     publisher = {mathdoc},
     volume = {431},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_431_a9/}
}
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M. V. Platonova. Nonprobabilistic infinitely divisible distributions: the L\'evy--Khinchin representation, limit theorems. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 145-177. http://geodesic.mathdoc.fr/item/ZNSL_2014_431_a9/