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Precise Asymptotics in the Law of Logarithm for U-statistics

Zhou Hui Lin Zhengyan (Department of Mathematics,Zhejiang University,Hangzhou 310027)  
Let {X_n;n≥1} be a sequence of i.i.d.random variables,and U_n be a U-statistic based on the symmetric kernel function h(x,y).Set U_n = 2/(n(n-1))(?) h(X_i,X_j),h_1(x) = Eh(x,X_2).Under some proper conditions,the exact moment convergence rates of sum from n=2 to∞(log n)~(δ-1) EU_n~2I{|U_n|≥n~(-1/2) log nε~(1/2)} and sum from n=3 to∞((log log n)~(δ-1))/(log n )EU_n~2I{|U_n|≥n~(-1/2) log log nε~(1/2)} are showed.
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