{"title":"On the Discrepancy of Random Walks on the Circle","authors":"A. Bazarova, I. Berkes, M. Raseta","doi":"10.2478/udt-2019-0015","DOIUrl":null,"url":null,"abstract":"Abstract Let X1,X2,... be i.i.d. absolutely continuous random variables, let Sk=∑j=1kXj {S_k} = \\sum\\nolimits_{j = 1}^k {{X_j}} (mod 1) and let D*N denote the star discrepancy of the sequence (Sk)1≤k≤N. We determine the limit distribution of ND N* \\sqrt N D_N^* and the weak limit of the sequence N(FN(t)-t) \\sqrt N \\left( {{F_N}(t) - t} \\right) in the Skorohod space D[0, 1], where FN (t) denotes the empirical distribution function of the sequence (Sk)1≤k≤N.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"43 1","pages":"73 - 86"},"PeriodicalIF":0.0000,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Uniform distribution theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/udt-2019-0015","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
Abstract Let X1,X2,... be i.i.d. absolutely continuous random variables, let Sk=∑j=1kXj {S_k} = \sum\nolimits_{j = 1}^k {{X_j}} (mod 1) and let D*N denote the star discrepancy of the sequence (Sk)1≤k≤N. We determine the limit distribution of ND N* \sqrt N D_N^* and the weak limit of the sequence N(FN(t)-t) \sqrt N \left( {{F_N}(t) - t} \right) in the Skorohod space D[0, 1], where FN (t) denotes the empirical distribution function of the sequence (Sk)1≤k≤N.