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

Benjamin Arras,Christian Houdré

doi:10.1214/19-ejp285

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.

Highlights

Stein’s method is a powerful device to quantify proximity in law between random variables
It has proven to be useful to compute explicit rates of convergence in several limiting results of probability theory
[21] proposes a general Stein’s method framework for target probability measures μ on Rd, d ≥ 1, which satisfy the following set of assumptions: μ has finite mean, is absolutely continuous with respect to the d-dimensional Lebesgue measure and its density is continuously differentiable with support the whole of Rd

Summary

Introduction

Stein’s method is a powerful device to quantify proximity in law between random variables. [21] proposes a general Stein’s method framework for target probability measures μ on Rd, d ≥ 1, which satisfy the following set of assumptions: μ has finite mean, is absolutely continuous with respect to the d-dimensional Lebesgue measure and its density is continuously differentiable with support the whole of Rd. Below, we introduce and develop a multidimensional Stein’s methodology for a different specific class of probability measures on Rd, namely non-degenerate selfdecomposable laws with finite first moment (see (2.4) in Section 2 for a definition). Thanks to earlier work on characterizing functionals of infinitely divisible distributions [14], we introduce in the last section of the present manuscript the relevant variational setting which ensures the existence of Stein kernels and implies manageable upper bounds on the Stein discrepancy.

Notations

Stein’s equation for SD laws via semigroup methods

Stein kernels for SD laws with finite second moment

A Appendix