## Electronic Journal of Probability

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

#### 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
Accepted: 2 March 2019
First available in Project Euclid: 26 March 2019

https://projecteuclid.org/euclid.ejp/1553565780

Digital Object Identifier
doi:10.1214/19-EJP285

Mathematical Reviews number (MathSciNet)
MR3933208

Zentralblatt MATH identifier
07055667

#### 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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