Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Bounds on the Poincaré constant for convolution measures

Thomas A. Courtade

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Abstract

We establish a Shearer-type inequality for the Poincaré constant, showing that the Poincaré constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincaré constants corresponding to convolutions of subsets of measures. This implies, for example, that the sequence of Poincaré constants corresponding to successive convolutions in the central limit theorem is non-increasing. We also establish a dimension-free stability estimate for subadditivity of the Poincaré constant on convolutions which uniformly improves an earlier one-dimensional estimate of a similar nature by Johnson (Teor. Veroyatn. Primen. 48 (2003) 615–620). As a byproduct of our arguments, we find that the various monotone properties of entropy, Fisher information and the Poincaré constant on convolutions have a common, simple root in Shearer’s inequality.

Résumé

Nous démontrons une inégalité de type Shearer pour les constantes de Poincaré, selon laquelle la constante correspondant à la convolution d’une famille de mesures peut être contrôlée de manière non-triviale par celles de convolutions de sous-familles. Ceci implique, par exemple, que les constantes de Poincaré décroissent de manière monotone le long du théorème central limite. Nous démontrons également une estimée de stabilité indépendante de la dimension pour la sous additivité des constantes de Poincaré de convolutions, améliorant un résultat unidimensionnel similaire dû à Johnson (Teor. Veroyatn. Primen. 48 (2003) 615–620). Comme conséquence de nos arguments, nous montrons que les diverses propriétés de monotonie de l’entropie, de l’information de Fisher et de la constantes de Poincaré pour les convolutions trouvent une même source en l’inégalité de Shearer.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 1 (2020), 566-579.

Dates
Received: 21 September 2018
Revised: 14 January 2019
Accepted: 13 February 2019
First available in Project Euclid: 3 February 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1580720501

Digital Object Identifier
doi:10.1214/19-AIHP973

Mathematical Reviews number (MathSciNet)
MR4059000

Zentralblatt MATH identifier
07199317

Subjects
Primary: 60E15: Inequalities; stochastic orderings 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Functional inequalities Poincaré inequalities Stability Convolution measures

Citation

Courtade, Thomas A. Bounds on the Poincaré constant for convolution measures. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 1, 566--579. doi:10.1214/19-AIHP973. https://projecteuclid.org/euclid.aihp/1580720501


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