{"title":"松弛矩条件下非线性统计中心极限定理的收敛速度","authors":"Nguyen Tien Dung","doi":"10.1007/s40306-021-00453-y","DOIUrl":null,"url":null,"abstract":"<div><p>This paper is concerned with normal approximation under relaxed moment conditions using Stein’s method. We obtain the explicit rates of convergence in the central limit theorem for (i) nonlinear statistics with finite absolute moment of order 2 + <i>δ</i> ∈ (2,3] and (ii) nonlinear statistics with vanishing third moment and finite absolute moment of order 3 + <i>δ</i> ∈ (3,4]. When applied to specific examples, these rates are of the optimal order <span>\\(O\\left (n^{-\\frac {\\delta }{2}}\\right )\\)</span> and <span>\\(O\\left (n^{-\\frac {1+\\delta }{2}}\\right )\\)</span>. Our proofs are based on the covariance identity formula and simple observations about the solution of Stein’s equation.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Rates of Convergence in the Central Limit Theorem for Nonlinear Statistics Under Relaxed Moment Conditions\",\"authors\":\"Nguyen Tien Dung\",\"doi\":\"10.1007/s40306-021-00453-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This paper is concerned with normal approximation under relaxed moment conditions using Stein’s method. We obtain the explicit rates of convergence in the central limit theorem for (i) nonlinear statistics with finite absolute moment of order 2 + <i>δ</i> ∈ (2,3] and (ii) nonlinear statistics with vanishing third moment and finite absolute moment of order 3 + <i>δ</i> ∈ (3,4]. When applied to specific examples, these rates are of the optimal order <span>\\\\(O\\\\left (n^{-\\\\frac {\\\\delta }{2}}\\\\right )\\\\)</span> and <span>\\\\(O\\\\left (n^{-\\\\frac {1+\\\\delta }{2}}\\\\right )\\\\)</span>. Our proofs are based on the covariance identity formula and simple observations about the solution of Stein’s equation.</p></div>\",\"PeriodicalId\":45527,\"journal\":{\"name\":\"Acta Mathematica Vietnamica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2021-10-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Vietnamica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40306-021-00453-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00453-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Rates of Convergence in the Central Limit Theorem for Nonlinear Statistics Under Relaxed Moment Conditions
This paper is concerned with normal approximation under relaxed moment conditions using Stein’s method. We obtain the explicit rates of convergence in the central limit theorem for (i) nonlinear statistics with finite absolute moment of order 2 + δ ∈ (2,3] and (ii) nonlinear statistics with vanishing third moment and finite absolute moment of order 3 + δ ∈ (3,4]. When applied to specific examples, these rates are of the optimal order \(O\left (n^{-\frac {\delta }{2}}\right )\) and \(O\left (n^{-\frac {1+\delta }{2}}\right )\). Our proofs are based on the covariance identity formula and simple observations about the solution of Stein’s equation.
期刊介绍:
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.