P. Eichelsbacher, Benedikt Rednoss, Christoph Thale, Guangqu Zheng
{"title":"离散情况下二阶高斯庞卡罗不等式的简化及其应用","authors":"P. Eichelsbacher, Benedikt Rednoss, Christoph Thale, Guangqu Zheng","doi":"10.1214/22-AIHP1247","DOIUrl":null,"url":null,"abstract":". 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.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"5 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"A simplified second-order Gaussian Poincaré inequality in discrete setting with applications\",\"authors\":\"P. Eichelsbacher, Benedikt Rednoss, Christoph Thale, Guangqu Zheng\",\"doi\":\"10.1214/22-AIHP1247\",\"DOIUrl\":null,\"url\":null,\"abstract\":\". 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.\",\"PeriodicalId\":42884,\"journal\":{\"name\":\"Annales de l Institut Henri Poincare D\",\"volume\":\"5 1\",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2021-08-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales de l Institut Henri Poincare D\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/22-AIHP1247\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-AIHP1247","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
A simplified second-order Gaussian Poincaré inequality in discrete setting with applications
. 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.