{"title":"On Proximity of Distributions of Successive Sums with Respect to the Prokhorov Distance","authors":"A. Yu. Zaitsev","doi":"10.1137/s0040585x97t991878","DOIUrl":null,"url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 217-226, August 2024. <br/> Let $X, X_1,\\dots, X_n,\\dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Let $F_{(n)}$ be the distribution of the normalized random vector $X/\\sqrt{n}$. Then $(X_1+\\dots+X_n)/\\sqrt{n}$ has distribution $F_{(n)}^n$ (the power is understood in the convolution sense). Let $\\pi(\\,{\\cdot}\\,,{\\cdot}\\,)$ be the Prokhorov distance. We show that, for any $d$-dimensional distribution $F$, there exist $c_1(F)>0$ and $c_2(F)>0$ depending only on $F$ such that $\\pi(F_{(n)}^n, F_{(n)}^{n+1})\\leqslant c_1(F)/\\sqrt n$ and $(F^n)\\{A\\} \\le (F^{n+1})\\{A^{c_2(F)}\\}+c_2(F)/\\sqrt{n}$, $(F^{n+1})\\{A\\} \\leq (F^n)\\{A^{c_2(F)}\\}+c_2(F)/\\sqrt{n}$ for each Borel set $A$ and for all natural numbers $n$ (here, $A^{\\varepsilon}$ denotes the $\\varepsilon$-neighborhood of a set $A$).","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":"7 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/s0040585x97t991878","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 217-226, August 2024. Let $X, X_1,\dots, X_n,\dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Let $F_{(n)}$ be the distribution of the normalized random vector $X/\sqrt{n}$. Then $(X_1+\dots+X_n)/\sqrt{n}$ has distribution $F_{(n)}^n$ (the power is understood in the convolution sense). Let $\pi(\,{\cdot}\,,{\cdot}\,)$ be the Prokhorov distance. We show that, for any $d$-dimensional distribution $F$, there exist $c_1(F)>0$ and $c_2(F)>0$ depending only on $F$ such that $\pi(F_{(n)}^n, F_{(n)}^{n+1})\leqslant c_1(F)/\sqrt n$ and $(F^n)\{A\} \le (F^{n+1})\{A^{c_2(F)}\}+c_2(F)/\sqrt{n}$, $(F^{n+1})\{A\} \leq (F^n)\{A^{c_2(F)}\}+c_2(F)/\sqrt{n}$ for each Borel set $A$ and for all natural numbers $n$ (here, $A^{\varepsilon}$ denotes the $\varepsilon$-neighborhood of a set $A$).
期刊介绍:
Theory of Probability and Its Applications (TVP) accepts original articles and communications on the theory of probability, general problems of mathematical statistics, and applications of the theory of probability to natural science and technology. Articles of the latter type will be accepted only if the mathematical methods applied are essentially new.