On Stable Random Variables with a Complex Stability Index

I. A. Alexeev^a^,*

^aSt. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, 191023 Russia

Correspondence to: *e-mail: vanyalexeev@list.ru

Received 31 August, 2021

Abstract—In this paper, we construct complex-valued random variables that satisfy the usual stability condition, but for a complex stability index α satisfying the conditions ${\text{|}}\alpha - 1{\text{|}} < 1$ and $\left| {\alpha - \frac{1}{2}} \right| \ne \frac{1}{2}$. 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.

Keywords: stable distributions, infinitely divisible distributions, limit theorems

DOI: 10.1134/S1064562421060028