U-统计量的精致渐近性

Precise Asymptotics of U-Statistics

设{X_n．n≥1}是一非退化的i．i．d．随机变量序列,U_n是以二维Borel可测对称函数h(x,y)为核函数的U-统计量．记U_n=2/(n(n-1))Σ_≤i≤j≤nh(X_i,X_j)．本文分别在核函数h(x,y)只有4/3阶矩或4/3+δ,0＜δ≤1的情况下,对非常广泛的一类权函数(x)与边界函数b(x)得到了如下关于U-统计量U_n的精致渐近性:不仅使得已有的结果成为我们的特况,还大大降低了其中的矩条件．

Let {X,Xn,n≥1} be non-degenerate i.i.d.random variables,define a U-statistics U_n=2/（n（n-1））∑≤j≤j≤nh（Xi,Xj）,where h（x,y） is a 2-dimensional Borel measurable and symmetric kernel function.Under the condition of E｜h（X1,X2）｜^4/3 or E｜h（X1,X2）｜^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 Un as follows： limε→αR（ε）∑n≥noφ（n）P（｜Un｜≥εn-1/2b（n））=1where 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.