Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Existence of Stein kernels under a spectral gap, and discrepancy bounds

Thomas A. Courtade, Max Fathi, and Ashwin Pananjady

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We establish existence of Stein kernels for probability measures on $\mathbb{R}^{d}$ satisfying a Poincaré inequality, and obtain bounds on the Stein discrepancy of such measures. Applications to quantitative central limit theorems are discussed, including a new central limit theorem in the Kantorovich–Wasserstein distance $W_{2}$ with optimal rate and dependence on the dimension. As a byproduct, we obtain a stable version of an estimate of the Poincaré constant of probability measures under a second moment constraint. The results extend more generally to the setting of converse weighted Poincaré inequalities. The proof is based on simple arguments of functional analysis.

Further, we establish two general properties enjoyed by the Stein discrepancy, holding whenever a Stein kernel exists: Stein discrepancy is strictly decreasing along the CLT, and it controls the third moments of a random vector.


Nous prouvons l’existence de noyaux de Stein pour les mesures de probabilités sur $\mathbb{R}^{d}$ satisfaisant une inégalité de Poincaré, et obtenons des bornes sur la discrépance de Stein de telles mesures. Des applications au théorème central limite sont données, dont une nouvelle borne sur la vitesse de convergence en distance de Kantorovitch–Wasserstein $W_{2}$ avec un taux et une dépendance en la dimension optimales. Comme corollaire, nous obtenons une version quantitative d’une borne sur la constante de Poincaré de mesures de probabilités satisfaisant une contrainte sur le moment d’ordre 2. Les résultats sont plus généralement valides dans le cadre de mesures vérifiant une inégalité de Poincaré à poids inversée. La preuve est basée sur des arguments simples d’analyse fonctionnelle.

De plus, nous démontrons deux propriétés générales sur la discrépance de Stein, valide dès lors qu’un noyau de Stein existe : la discrépance de Stein est strictement décroissante le long du TCL, et elle contrôle le moment d’ordre 3 d’un vecteur aléatoire.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 777-790.

Received: 14 April 2017
Revised: 26 February 2018
Accepted: 15 March 2018
First available in Project Euclid: 14 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60B10: Convergence of probability measures 60E15: Inequalities; stochastic orderings

Stein kernels Quantitative central limit theorems Poincaré inequalities


Courtade, Thomas A.; Fathi, Max; Pananjady, Ashwin. Existence of Stein kernels under a spectral gap, and discrepancy bounds. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 777--790. doi:10.1214/18-AIHP898.

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