Electronic Journal of Probability

On Stein’s method for multivariate self-decomposable laws

Benjamin Arras and Christian Houdré

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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, Poincaré-type inequalities for self-decomposable laws having finite first moment are revisited. In this non-local setting, several algebraic quantities (such as the carré du champs and its iterates) originating in the theory of Markov diffusion operators are computed. Finally, rigidity and stability results for the Poincaré-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 $\alpha \to 2$.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 128, 63 pp.

Received: 14 August 2019
Accepted: 16 October 2019
First available in Project Euclid: 9 November 2019

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Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms 60F05: Central limit and other weak theorems

infinite divisibility self-decomposability Stein’s method stable laws Dirichlet forms smooth Wassertein distance Stein’s kernel integro-differential equations Poincaré inequality

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Arras, Benjamin; Houdré, Christian. On Stein’s method for multivariate self-decomposable laws. Electron. J. Probab. 24 (2019), paper no. 128, 63 pp. doi:10.1214/19-EJP378. https://projecteuclid.org/euclid.ejp/1573268592

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