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

Abstract

Let {X, X-n; n >= 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 < infinity, where h is a measurable symmetric function from S-m. into R = (-infinity, infinity). Let {w(n,i1,i2, ..., im); 1 <= i(1) < i(2) < ... < i(m) <= 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 [GRAPHICS] whenever sup(n >= m) max(1 <= i1 < i2 < ... < im <= n) vertical bar w(n,i1,i2, ..., im)vertical bar < 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.