Electronic Journal of Probability

On Stein’s method for multivariate self-decomposable laws with finite first moment

Benjamin Arras and Christian Houdré

Full-text: Open access

Abstract

We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R} ^d$ having finite first moment. Building on previous univariate findings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy specifically designed for infinitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\mathbb{R} ^d$ and a non-degenerate self-decomposable target law with finite second moment. Finally, under an appropriate Poincaré-type inequality assumption, we investigate, via variational methods, the existence of Stein kernels. In particular, this leads to quantitative versions of classical results on characterizations of probability distributions by variational functionals.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 29, 33 pp.

Dates
Received: 4 October 2018
Accepted: 2 March 2019
First available in Project Euclid: 26 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1553565780

Digital Object Identifier
doi:10.1214/19-EJP285

Mathematical Reviews number (MathSciNet)
MR3933208

Zentralblatt MATH identifier
07055667

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

Keywords
infinite divisibility self-decomposability Stein’s method Stein’s kernel weak limit theorems rates of convergence smooth Wassertein distance

Rights
Creative Commons Attribution 4.0 International License.

Citation

Arras, Benjamin; Houdré, Christian. On Stein’s method for multivariate self-decomposable laws with finite first moment. Electron. J. Probab. 24 (2019), paper no. 29, 33 pp. doi:10.1214/19-EJP285. https://projecteuclid.org/euclid.ejp/1553565780


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