A unified approach to Stein’s method for stable distributions

In this article, we propose a modified technique for finding Stein operator for the class of infinitely divisible distributions using its characteristic function that relaxes the assumption of the first finite moment. Using this technique, we reproduce the Stein operators for stable distributions with $\alpha\in(0,2)$ with less efforts. We have shown that a single approach with minor modifications is enough to solve the Stein equations for the stable distributions with $\alpha\in(0,1)$ and $\alpha\in(1,2)$. Finally, we give applications of our results for stable approximations.