A Remark on Strong Law of Large Numbers for Weighted U-Statistics  

作　　者：Hyung-Tae HA Mei Ling HUANG 

机构地区：[1]Department of Applied Statistics,Gachon University [2]Department of Mathematics and Statistics,Brock University

出　　处：《Acta Mathematica Sinica,English Series》2014年第9期1595-1605,共11页数学学报（英文版）

基　　金：The first author is supported by Basic Science Research Program through the National Research Foundationof Korea funded by the Ministry of Education,Science,and Technology(Grant No.2011-0013791);the secondauthor is partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada;the third author is partially supported by a grant from the Natural Sciences and Engineering Research Councilof Canada

摘　　要：Let （X, Xn; n≥ 1} be a sequence of i.i.d, random variables with values in a measurable space （S,8） such that E｜h（X1, X2,..., Xm）｜ 〈 ∞, where h is a measurable symmetric function from Sm into R = （-∞, ∞）. Let {wn,i1,i2 im ; 1 ≤ i1 〈 i2 〈 …… 〈im 〈 n, n ≥ m} be a matrix array of real numbers. Motivated by a result of Choi and Sung （1987）, in this note we are concerned with establishing a strong law of large numbers for weighted U-statistics with kernel h of degree m. We show that whenever SUP n≥m max1〈i1〈i2〈…〈im≤｜wn i1,i2 i,im｜ 〈∞, where 0 = Eh（X1, X2,..., Xm）. The proof of this result is based on a new general result on complete convergence, which is a fundamental tool, for array of real-valued random variables under some mild conditions.Let （X, Xn; n≥ 1} be a sequence of i.i.d, random variables with values in a measurable space （S,8） such that E｜h（X1, X2,..., Xm）｜ 〈 ∞, where h is a measurable symmetric function from Sm into R = （-∞, ∞）. Let {wn,i1,i2 im ; 1 ≤ i1 〈 i2 〈 …… 〈im 〈 n, n ≥ m} be a matrix array of real numbers. Motivated by a result of Choi and Sung （1987）, in this note we are concerned with establishing a strong law of large numbers for weighted U-statistics with kernel h of degree m. We show that whenever SUP n≥m max1〈i1〈i2〈…〈im≤｜wn i1,i2 i,im｜ 〈∞, where 0 = Eh（X1, X2,..., Xm）. The proof of this result is based on a new general result on complete convergence, which is a fundamental tool, for array of real-valued random variables under some mild conditions.

关 键 词：Strong law of large numbers weighted U-statistics complete convergence 

分 类 号：O211.4[理学—概率论与数理统计]