Strong representation of weak convergence  

作　　者：HU Jiang BAI ZhiDong 

机构地区：[1]Key Laboratory of Applied Statistics of Ministry of Education and School of Mathematics & Statistics,Northeast Normal University

出　　处：《Science China Mathematics》2014年第11期2399-2406,共8页中国科学：数学（英文版）

基　　金：supported by the Fundamental Research Funds for the Central Universities;Program for Changjiang Scholars and Innovative Research Team in University;National Natural Science Foundation of China(Grant Nos.11301063 and 11171057)

摘　　要：Skorokhod's representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→μ0,as n →∞,then there exist a probability space(Ω,F,P) and a sequence of random elements Xnsuch that Xn→ X almost surely and Xnhas the distribution function μn,n = 0,1,2,... We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces Sn,a sequence of probability measures μnand a sequence of measurable mappings n such that μnn-1w→μ0,then there exist a probability space(Ω,F,P) and Sn-valued random elements Xndefined on Ω,with distribution μnand such that n(Xn) → X0 almost surely. In addition,we present several applications of our result including some results in random matrix theory,while the original Skorokhod representation theorem is not applicable.Skorokhod＇s representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→ μ0,as n → ∞,then there exist a probability space（Ω,F,P） and a sequence of random elements Xnsuch that Xn→ X almost surely and Xnhas the distribution function μn,n = 0,1,2,... We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces Sn,a sequence of probability measures μnand a sequence of measurable mappings n such that μnn-1w→ μ0,then there exist a probability space（Ω,F,P） and Sn-valued random elements Xndefined on Ω,with distribution μnand such that n（Xn） → X0 almost surely. In addition,we present several applications of our result including some results in random matrix theory,while the original Skorokhod representation theorem is not applicable.

关 键 词：Skorohod＇s representation theorem strong representation of weak convergence random matrices 

分 类 号：O211.4[理学—概率论与数理统计] O189.11[理学—数学]