{"title":"Some remarks on the ergodic theorem for U-statistics","authors":"Herold G. Dehling, Davide Giraudo, Dalibor Volny","doi":"10.5802/crmath.494","DOIUrl":null,"url":null,"abstract":"In this note, we investigate the convergence of a U-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and L 1 convergence and present some counter-examples showing that the U-statistic itself might fail to converge: centering is needed as well as finiteness of sup j≥2 𝔼[|h(X 1 ,X j )|].","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"83 7","pages":"0"},"PeriodicalIF":0.8000,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Comptes Rendus Mathematique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/crmath.494","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this note, we investigate the convergence of a U-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and L 1 convergence and present some counter-examples showing that the U-statistic itself might fail to converge: centering is needed as well as finiteness of sup j≥2 𝔼[|h(X 1 ,X j )|].
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The Comptes Rendus - Mathématique cover all fields of the discipline: Logic, Combinatorics, Number Theory, Group Theory, Mathematical Analysis, (Partial) Differential Equations, Geometry, Topology, Dynamical systems, Mathematical Physics, Mathematical Problems in Mechanics, Signal Theory, Mathematical Economics, …
Articles are original notes that briefly describe an important discovery or result. The articles are written in French or English.
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