A unified approach to Stein’s method for stable distributions

Abstract

In this article, we first review the connection between Lévy processes and infinitely divisible random variables, and discuss the classification of infinitely divisible distributions. Next, we establish a Stein identity for an infinitely divisible random variable via the Lévy-Khintchine representation of the characteristic function. In particular, we establish and unify the Stein identities for an α-stable random variable available in the existing literature. Next, we derive the solutions of the Stein equations and its regularity estimates. Further, we derive error bounds for α-stable approximations, and also obtain rates of convergence results in Wasserstein-$\mathit{\delta },\mathit{\delta }<\mathit{\alpha }$ for $\mathit{\alpha }\in (0,1)$ and Wasserstein-type distances for $\mathit{\alpha }\in (1,2)$. Finally, we compare these results with existing literature.