A Unified Approach to Stein's Method for Stable Distributions

Abstract: In this article, we first review the connection between L\'evy processes and infinitely divisible random variables, and the classification of infinitely divisible distributions. Using this connection and the L\'evy-Khinchine representation of the characteristic function, we establish a Stein identity for an infinitely divisible random variable. The classification and slight modification in approach give us a Stein identity for an $\alpha$-stable random variable with $\alpha\in (0,2).$ Using fine regularity estimates for the solution to Stein equation, we derive error bounds for $\alpha$-stable approximations. We then apply these results to obtain rates of convergence. Finally, we compare these rates with the results available in the literature.