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

Let $\lbrace X_n, n \ge 1 \rbrace$ 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 $\sigma_f$, where $\sigma_f^2 = \underset{n \to \infty}{lim}(Var S_n)/n$, $S_n = \sum_{j=1}^n f(X_j)$, and $f$ is a non-monotonic function, using the ideas of Peligrad and Suresh $(1995)$.