Let {X,Xn; n ≥ 1} be a sequence of i.i.d. random variables with values in a measurable space \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$(\mathbb{S},\mathcal{S})$\end{document} such that \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mathbb{E}|h(X_1 ,X_2 ,...,X_m )| < \infty $\end{document}, where h is a measurable symmetric function from \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mathbb{S}^m $\end{document} into ℝ = (−∞,∞). Let \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\{ w_{n,i_1 ,i_2 ,...i_m } ;1 \leqslant i_1 < i_2 < \cdots i_m \leqslant n,n \geqslant m\} $\end{document} be a matrix array of real numbers. Motivated by a result of Choi and Sung (1987), in this note we are concerned with establishing a strong law of large numbers for weighted U-statistics with kernel h of degree m. We show that \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mathop {\lim }\limits_{n \to \infty } \frac{{m!(n - m)!}}
{{n!}}\sum\limits_{1 \leqslant i_1 < i_2 < \cdots i_m \leqslant n} {w_{n,i_1 ,i_2 ,...,i_m } (h(X_{i_1 } ,X_{i_2 } ,...,X_{i_m } ) - \theta ) = 0} a.s.
$\end{document} whenever supn≥m\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\max _{1 \leqslant i_1 < i_2 < \cdots i_m \leqslant n} |w_{n,i_1 ,i_2 , \cdots ,i_m } | < \infty $\end{document} where \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\theta = \mathbb{E}h(X_1 ,X_2 ,...,X_m )$\end{document}. 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.
