Hsu-Robbins Type Theorems for Pairwise Products Sums of Ｉ.Ｉ.D. Random Variables (I)

Let X1, X2,... be independent identically distributed (iid) random variables, Un = 1≤i＜j≤nXiXj is said to be pairwise products sum. In this paper, we give necessary and sufficient conditions for Un/n~2 → 0 (i.e. (0.3)). Though this question in this part has not been settled thoroughly yet, it reveals an essential distinction between this class conditions and these ones of Sn/n → 0 (Sn = n=1 Xi), i.e. this class conditions can not been characterized thoroughly by the moment of X1, it need to depend deeply on the tail probabality property of X1.