On Stein’s method for multivariate self-decomposable laws

Abstract

This work explores and develops elements of Stein's method of approximation, in the infinitely divisible setting, and its connections to functional analysis. It is mainly concerned with multivariate self-decomposable laws without finite first moment and, in particular, with α-stable ones, α ∈ (0, 1]. At first, several characterizations of these laws via covariance identities are presented. In turn, these characterizations lead to integro-differential equations which are solved with the help of both semigroup and Fourier methodologies. Then, Poincaré-type inequalities for self-decomposable laws having finite first moment are revisited. In this non-local setting, several algebraic quantities (such as the carré du champs and its iterates) originating in the theory of Markov diffusion operators are computed. Finally, rigidity and stability results for the Poincaré-ratio functional of the rotationally invariant α-stable laws, α ∈ (1, 2), are obtained; and as such they recover the classical Gaussian setting as α → 2.

Covariance Representations, $L^p$-Poincaré Inequalities, Stein's Kernels and High Dimensional CLTs

“…The semigroup (1.15) is the classical Gaussian Ornstein-Uhlenbeck semigroup and the semigroup (1.16) is the Ornstein-Uhlenbeck semigroup associated with the α-stable probability measure µ α and recently put forward in the context of Stein's method for self-decomposable distributions (see [6,7,8]). Finally, denoting by ((P να t ) * ) t≥0 the formal adjoint of the semigroup (P να t ) t≥0 , the "carré de Mehler" semigroup is defined, for all t ≥ 0, by…”

“…In the context of Stein's method for self-decomposable laws, a general covariance identity has been obtained in [8,Theorem 5.10] in the framework of closed symmetric non-negative definite bilinear forms with dense domain under some coercive assumption. Indeed, the identity (5.15) there can be understood as a generalization of (2.5) in [28] and of (3.2) in [42] from which asymmetric covariance estimates can be obtained.…”

Estimation of smooth functionals in high-dimensional models: bootstrap chains and Gaussian approximation

“…In recent papers [15,20], another approach to high-dimensional normal approximation has been developed It is based on the technique of Stein kernels and it applies to probability distributions in R d with bounded Poincaré constants, in particular, to some log-concave distributions (see also [3] for more general results).…”

“…where A(θ) : E → E is a bounded linear operator, {η j } are independent r.v. with 3 Eη j = 0, Eη 2 j = 1, j = 1, . .…”

Stein's method for Conditional Central Limit Theorem

“…State of the art in establishing CLT and getting a convergence rate is Stein's method, which we will discuss in further detail in Section 1.2. Researchers have applied Stein's method to prove and find a rate of convergence in Multivariate CLT [3,11,5,25,26,35,38], Martingale CLT [39], Local Limit theorem [5,36], and in other non-Gaussian limit theorems (see [1,10,12,13,31] among many others). Stein's method has also been applied to prove concentration inequalities [7,9,21,23], moderate deviation results [14], and strong coupling [8].…”