关键词: U-统计量, 对数律, 重对数律, 精确渐近性, 矩完全收敛性

Abstract:

Let {X_n; n ≥1} be a sequence of i.i.d. random variables, and Un be a U-statistic based on the symmetric kernel function h(x, y). Set U_n =2/n(n-1) ∑_1≤i<j≤n h(X_i, X_j), h₁(x) =Eh(x, X₂). Under someproperconditions, the exactmomentconvergenceratesof ∑_n=2 ^∞(logn)^δ-1EU_n²I {I U _n |≥n ^1/2√lognε} and ∑_n=3^∞(loglognε)^δ-1/logn EU_n² I {|U _n |≥n^1/2 √log lognε } are showed.

Key words: U-statistics, Law of the iterated logarithm, Precise asymptotics, Complete mo-ment convergence