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On limiting distribution of U-statistics based on associated random variables

Abstract

Let {Xn,n1}\lbrace X_n, n \ge 1 \rbrace be a sequence of stationary associated random variables. Based on this sample, we establish a central limit theorem for U-statistics with monotonic kernels of degree 3 and above using the Hoeffding's decomposition. We also extend these results to U-statistics based on non-monotonic functions. We also obtain a consistent estimator of σf\sigma_f, where σf2=limn(VarSn)/n\sigma_f^2 = \underset{n \to \infty}{lim}(Var S_n)/n, Sn=j=1nf(Xj)S_n = \sum_{j=1}^n f(X_j), and ff is a non-monotonic function, using the ideas of Peligrad and Suresh (1995)(1995).

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