February 2023 A simplified second-order Gaussian Poincaré inequality in discrete setting with applications
Peter Eichelsbacher, Benedikt Rednoß, Christoph Thäle, Guangqu Zheng
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 59(1): 271-302 (February 2023). DOI: 10.1214/22-AIHP1247


In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin–Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdős–Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial–Meshulam–Wallach random κ-complex and infinite weighted 2-runs are treated.

Dans cet article, une inégalité de Poincaré gaussienne simplifiée du second ordre pour l’approximation normale de fonctionnelles sur une infinité de variables aléatoires de Rademacher est obtenue. Elle est basée sur une nouvelle limite pour la distance de Kolmogorov entre une fonction générale de Rademacher et une variable gaussienne, qui est établie au moyen de la méthode discrète de Malliavin–Stein et qui présente un intérêt indépendant. Comme application, le nombre de sommets de degré prescrit et la statistique de comptage des sous-graphes dans le graphe aléatoire d’Erdős–Rényi sont discutés. Le nombre de sommets de degré fixé est également étudié pour la percolation sur l’hypercube de Hamming. De plus, le nombre de faces isolées dans le κ-complexe aléatoire de Linial–Meshulam–Wallach et les succès consécutifs pondérés infinis sont examinés.

Funding Statement

BR has been supported by the German Research Foundation (DFG) under project number 459731056.
CT has been supported by the DFG priority program SPP 2265 Random Geometric Systems.


Download Citation

Peter Eichelsbacher. Benedikt Rednoß. Christoph Thäle. Guangqu Zheng. "A simplified second-order Gaussian Poincaré inequality in discrete setting with applications." Ann. Inst. H. Poincaré Probab. Statist. 59 (1) 271 - 302, February 2023. https://doi.org/10.1214/22-AIHP1247


Received: 6 September 2021; Revised: 4 January 2022; Accepted: 11 January 2022; Published: February 2023
First available in Project Euclid: 16 January 2023

MathSciNet: MR4533729
Digital Object Identifier: 10.1214/22-AIHP1247

Primary: 05C80 , 60F05 , 60H07

Keywords: Berry–Esseen bound , Discrete stochastic analysis , Erdős–Rényi random graph , Hypercube percolation , Infinite weighted 2-run , Isolated face , Malliavin–Stein method , Rademacher functional , random simplicial complex , second-order Poincaré inequality , subgraph count , Vertex of given degree

Rights: Copyright © 2023 Association des Publications de l’Institut Henri Poincaré


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Vol.59 • No. 1 • February 2023
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