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

作者：Ha Hyung Tae; Huang Mei Ling; Li De Li^*

来源：Acta Mathematica Sinica-English Series, 2014, 30(9): 1595-1605.

DOI：10.1007/s10114-014-1601-5

• Strong law of large numbers
• weighted U-statistics
• complete convergence

Let {X, X-n; n %26gt;= 1} be a sequence of i.i.d. random variables with values in a measurable space (5, S) such that E vertical bar h(X-1, X-2, ..., X-m)vertical bar %26lt; infinity, where h is a measurable symmetric function from S-m. into R = (-infinity, infinity). Let {w(n,i1,i2, ..., im); 1 %26lt;= i(1) %26lt; i(2) %26lt; ... %26lt; i(m) %26lt;= n, n %26gt;= 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 %26lt;br%26gt;[GRAPHICS] %26lt;br%26gt;whenever sup(n %26gt;= m) max(1 %26lt;= i1 %26lt; i2 %26lt; ... %26lt; im %26lt;= n) vertical bar w(n,i1,i2, ..., im)vertical bar %26lt; infinity, where theta = Eh(X-1, X-2, ..., X-m). 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.

• 出版日期2014-9