Ji Gao YAN(1); Chun SU(2)
Let {X,X_n,n≥1} be non-degenerate i.i.d.random variables,define a U-statistics U_n=2/(n(n-1))∑_≤j≤j≤nh(X_i,X_j),where h(x,y) is a 2-dimensional Borel measurable and symmetric kernel function.Under the condition of E|h(X_1,X_2)|~(4/3) or E|h(X_1,X_2)|~((4/3)+δ),0<δ≤1 separately,for a very extensive weighted functionφ(x) and a boundary function b(x),we discuss the precise asymptotics of U_n as follows: where a is an appropriate non-negative number.It not only makes the known results on this subject as the special case of our results,but also reduces their moment conditions.