Stable random variables with complex stability index, II

This paper, which is a continuation of [I. A. Alexeev, Theory Probab. Appl., 67 (2022), pp. 335–351], is concerned with $\alpha$-stable distributions with complex stability index $\alpha$. Sufficient conditions for membership in the domain of attraction of $\alpha$-stable random variables (r.v.'s) are given, and $\alpha$-stable Lévy processes and the corresponding semigroups of operators are constructed. Necessary and sufficient conditions are given for membership in the class of limit laws for sums of independent and identically distributed (i.i.d.) complex r.v.'s with complex normalization and centering.