On Stein's Method for Multivariate Self-Decomposable Laws

Benjamin Arras, Christian Houdré

Published 2019-07-23Version 1

In this work, we pursue the investigation and the development of Stein's method in the infinitely divisible setting and its relation with functional analysis. Section $2$ starts with standard notations and definitions together with a multidimensional characterization theorem for infinitely divisible distributions with finite first moment. Based on this result and on a truncation procedure, Section $3$ develops characterization results for multivariate self-decomposable laws without finite first moment highlighting the role of the L\'evy-Khintchine representation of the characteristic function of the target self-decomposable distribution. In particular, these results apply to multivariate stable laws with stability parameter belonging to $(0,1]$. In Section $4$, Stein's equation for self-decomposable distributions without finite first moment is set down and solved thanks to a combination of semigroup technics and Fourier analysis. Finally, in the last section of this note, we take a new look at Poincar\'e-type inequalities for self-decomposable laws with finite first moment. A proof based on semigroup and on Fourier analysis is presented. Several algebraic quantities from Markov diffusion operators theory are computed in this non-local setting and in particular for the rotationally invariant $\alpha$-stable laws with $\alpha \in (1,2)$. Finally, rigidity and stability results for the Poincar\'e $U$-functional of the rotationally invariant $\alpha$-stable distribution are obtained thanks to spectral analysis and Dirichlet form theory.