A remark on strong law of large numbers for weighted \(U\)-statistics (Q741244)

This paper establishes a strong law of large numbers for weighted \(U\)-statistics with kernel function of degree \(m\) based on independent and identically distributed random variables, provided that the weights are uniformly bounded. The result for the case \(m=1\) coincides with a result in [\textit{B. D. Choi} and \textit{S. H. Sung}, Stochastic Anal. Appl. 5, 365--377 (1987; Zbl 0633.60049)]. The proof presented in this paper utilizes a new general result on complete convergence of an array of real-valued random variables which is of independent interest.