{"title":"Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes","authors":"Marius Butzek, Peter Eichelsbacher","doi":"arxiv-2409.09439","DOIUrl":null,"url":null,"abstract":"In this paper we obtain non-uniform Berry-Esseen bounds for normal\napproximations by the Malliavin-Stein method. The techniques rely on a detailed\nanalysis of the solutions of Stein's equations and will be applied to\nfunctionals of a Gaussian process like multiple Wiener-It\\^o integrals, to\nPoisson functionals as well as to the Rademacher chaos expansion. Second-order\nPoincar\\'e inequalities for normal approximation of these functionals are\nconnected with non-uniform bounds as well. As applications, elements living\ninside a fixed Wiener chaos associated with an isonormal Gaussian process, like\nthe discretized version of the quadratic variation of a fractional Brownian\nmotion, are considered. Moreover we consider subgraph counts in random\ngeometric graphs as an example of Poisson $U$-statistics, as well as subgraph\ncounts in the Erd\\H{o}s-R\\'enyi random graph and infinite weighted 2-runs as\nexamples of functionals of Rademacher variables.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"209 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09439","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we obtain non-uniform Berry-Esseen bounds for normal
approximations by the Malliavin-Stein method. The techniques rely on a detailed
analysis of the solutions of Stein's equations and will be applied to
functionals of a Gaussian process like multiple Wiener-It\^o integrals, to
Poisson functionals as well as to the Rademacher chaos expansion. Second-order
Poincar\'e inequalities for normal approximation of these functionals are
connected with non-uniform bounds as well. As applications, elements living
inside a fixed Wiener chaos associated with an isonormal Gaussian process, like
the discretized version of the quadratic variation of a fractional Brownian
motion, are considered. Moreover we consider subgraph counts in random
geometric graphs as an example of Poisson $U$-statistics, as well as subgraph
counts in the Erd\H{o}s-R\'enyi random graph and infinite weighted 2-runs as
examples of functionals of Rademacher variables.