Asymptotic Normality of k-U Statistic

U_n^m, k=\sum_1 \leq i_1<\cdots<i_m \leq k g\left(X_i_1, \cdots, X_i_m\right)+\sum_i_m-k+1 \leq i_1<\cdots<i_m-1<i_m \sum_k+1 \leq i_m \leq n g\left(X_i_1, \cdots, X_i_m\right) is called the k-U statistic, where g\left(x_1, \cdots, x_m\right) is a symmetric function, k is a positive integer, and equal to or less than n and probably dependent on n. This statistic is a class of statistic and when k=n,\binomnm^-1 U_n^m, n is U-statistic. The asymptotic normality of the k-U statistic will be studied in this paper.