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 $(\mathbb{S},\mathcal{S})$ such that $\mathbb{E}|h(X_1 ,X_2 ,...,X_m )| < \infty$, where h is a measurable symmetric function from $\mathbb{S}^m$ into ℝ = (−∞,∞). Let $\{ w_{n,i_1 ,i_2 ,...i_m } ;1 \leqslant i_1 < i_2 < \cdots i_m \leqslant n,n \geqslant m\}$ 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

$\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. $

whenever sup _n≥m $\max _{1 \leqslant i_1 < i_2 < \cdots i_m \leqslant n} |w_{n,i_1 ,i_2 , \cdots ,i_m } | < \infty$ where $\theta = \mathbb{E}h(X_1 ,X_2 ,...,X_m )$. 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.