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 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 R d and a non-degenerate self-decomposable target law with finite second moment. Finally, under an appropriate spectral gap 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.

On Stein’s method for multivariate self-decomposable laws

“…Initially obtained in [17] (see also [32]), these Poincaré-type inequalities reflect the infinite divisibility of the reference measure (without Gaussian component) and as such put into play a non-local Dirichlet form contrasting with the standard local Dirichlet form associated with the Gaussian measures. Our new proof of these Poincaré-type inequalities is based on the semigroup of operators already put forward (and used to solve the Stein equation) in [2] and is in line with the proof of the Gaussian Poincaré inequality based on the differentiation of the variance along the Ornstein-Uhlenbeck semigroup (see, e.g., [6]). Moreover, in this non-local setting, we compute several algebraic quantities (such as the carré du champs and its iterates) originating in Markov diffusion operators theory in order to reach rigidity and stability results for the Poincaré-ratio (U -) functional defined in (70) and associated with the rotationally invariant α-stable distributions.…”

“…Moreover, g j − g R,j L p (µ) → 0, as R tends to +∞, for all p ≤ β. Since, (see [2,Proposition 3.4])…”

“…The present notes form a sequel to the works [1,2] where Stein's method for general univariate and multivariate infinitely divisible laws with finite first moment has been initiated. Introduced in [53,54], Stein's method is a collection of methods allowing to control the discrepancy, in a suitable metric, between probability measures and to provide quantitative rates of convergence in weak limit theorems.…”

“…The methodology developed here for multivariate self-decomposable distributions relies on a specific semigroup of operators already put forward in our previous analyses [1,2]. The generator of this semigroup is an integro-differential operator whose non-local part depends in a subtle way on the Lévy measure of the target self-decomposable distribution (see Lemma 4.1).…”

“…Moreover, this equation reflects the Lévy-Khintchine representation used to express the characteristic function of the target self-decomposable distribution. This naturally induces three types of equations reminiscent of the following classical distinction between stable laws: α ∈ (0, 1), α = 1 and α ∈ (1,2).…”