# On Stein's Method for Multivariate Self-Decomposable Laws

## TL;DR

This paper advances Stein's method for multivariate self-decomposable laws, especially $eta$-stable distributions, by providing new characterizations, integro-differential equations, and stability results, connecting probability, analysis, and functional inequalities.

## Contribution

It introduces novel characterizations and equations for multivariate self-decomposable laws, extending Stein's method to non-Gaussian stable distributions and analyzing their stability and inequalities.

## Key findings

- Characterizations of self-decomposable laws via covariance identities
- Solutions to integro-differential equations using semigroup and Fourier methods
- Stability results for $eta$-stable laws as $eta o 2$

## Abstract

This work explores and develops elements of Stein's method of approximation, in the infinitely divisible setting, and its connections to functional analysis. It is mainly concerned with multivariate self-decomposable laws without finite first moment and, in particular, with $\alpha$-stable ones, $\alpha \in (0,1]$. At first, several characterizations of these laws via covariance identities are presented. In turn, these characterizations lead to integro-differential equations which are solved with the help of both semigroup and Fourier methodologies. Then, Poincar\'e-type inequalities for self-decomposable laws having finite first moment are revisited. In this non-local setting, several algebraic quantities (such as the carr\'e du champs and its iterates) originating in the theory of Markov diffusion operators are computed. Finally, rigidity and stability results for the Poincar\'e-ratio functional of the rotationally invariant $\alpha$-stable laws, $\alpha\in (1,2)$, are obtained; and as such they recover the classical Gaussian setting as $\alpha \to 2$.