On stable random variables with a complex stability index

In this paper, we construct complex-valued random variables that satisfy the usual stability condition, but for a complex stability index $\alpha$ satisfying the conditions $|\alpha-1|<1$ and $|\alpha-\frac12|\ne\frac12$. A representation of the characteristic functions of the constructed random variables is found, and limit theorems for sums of independent identically distributed random variables are formulated.