Nonprobabilistic infinitely divisible distributions: the Lévy–Khinchin representation, limit theorems

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.