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2019 On Stein’s method for multivariate self-decomposable laws with finite first moment
Benjamin Arras, Christian Houdré
Electron. J. Probab. 24: 1-33 (2019). DOI: 10.1214/19-EJP285


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.


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Benjamin Arras. Christian Houdré. "On Stein’s method for multivariate self-decomposable laws with finite first moment." Electron. J. Probab. 24 1 - 33, 2019.


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

zbMATH: 07055667
MathSciNet: MR3933208
Digital Object Identifier: 10.1214/19-EJP285

Primary: 60E07 , 60E10 , 60F05

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


Vol.24 • 2019
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