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 （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.展开更多
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 Canadathe third author is partially supported by a grant from the Natural Sciences and Engineering Research Councilof Canada
文摘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.
