Let $\{X_1,\ldots,X_m \}$ and $\{Y_1, \ldots, Y_n \}$ be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functions $F$ and $G$, respectively. We study the properties of the classical Wilcoxon-Mann-Whitney statistic for testing for stochastic dominance in the above set up.