Kernkonzepte
This work provides new stability estimates for centered non-degenerate self-decomposable probability measures on Rd with finite second moment and for non-degenerate symmetric α-stable probability measures on Rd with α ∈ (1, 2). The proofs rely on Stein's method, closed forms techniques, and weighted Poincaré-type inequalities. The results yield explicit rates of convergence in Wasserstein-type distances for several instances of generalized central limit theorems.
Zusammenfassung
The content presents new stability estimates for two classes of probability measures on Rd: centered non-degenerate self-decomposable measures with finite second moment, and non-degenerate symmetric α-stable measures with α ∈ (1, 2).
For the self-decomposable case:
The stability estimate is obtained using Stein's method, closed forms techniques, and a new weighted Poincaré-type inequality.
The estimate provides an upper bound on the Wasserstein-2 distance between the target self-decomposable measure and a given probability measure, in terms of the moments of their difference and the Stein kernel.
For the symmetric α-stable case with α ∈ (1, 2):
The stability estimate is derived using the generator approach to Stein's method and a smooth truncation argument.
The estimate bounds the 1-Wasserstein distance between the target symmetric α-stable measure and a given probability measure, in terms of certain integrals involving the weight function and the α-stable Lévy measure.
As applications, the work derives sharp quantitative rates of convergence in Wasserstein-type distances for several instances of generalized central limit theorems with self-decomposable and symmetric stable limiting laws. In particular, a n^(1-2/α) rate is obtained in 1-Wasserstein distance when the target law is a non-degenerate symmetric α-stable one with α ∈ (1, 2).
The non-degenerate symmetric Cauchy (α = 1) case is also studied in depth from a spectral perspective, leading to a n^(-1) rate of convergence when the initial law is a certain instance of layered stable distributions.