Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods

In these notes, we obtain new stability estimates for centered non-degenerate selfdecomposable probability measures on $\mathbb{R}^d$ with finite second moment and for non-degenerate symmetric $\alpha$-stable probability measures on $\mathbb{R}^d$ with $\alpha \in [1,2)$. These new results are refinements of the corresponding ones available in the literature. The proofs are based on Stein's method for self-decomposable laws, recently developed in a series of papers, and on closed forms techniques together with a new ingredient: weighted Poincar\'e-type inequalities. As applications, rates of convergence in Wasserstein-type distances are computed for several instances of the generalized central limit theorems (CLTs). In particular, a $n^{1-2/\alpha}$-rate is obtained in $1$-Wasserstein distance when the target law is a non-degenerate symmetric $\alpha$-stable one with $\alpha \in (1,2)$. Finally, the non-degenerate symmetric Cauchy case is studied at length from a spectral point of view. At last, in this Cauchy situation, a $n^{-1}$-rate of convergence is obtained when the initial law is a certain instance of layered stable distributions.