" 

 :I removed the section on limit theorems.  I am not finding any support for discrete stable distributions satisfying this property except for lattice random variables.  In fact  explicitly states that at least some discrete stable distributions belong ""to the domain of normal attraction of the absolutely continuous positive stable distribution (stable distribution with index of stability α = γ and with skewness parameter β = 1), whose Laplace transform is given by exp(−λtγ ).""   "