Let be independent copies of a stationary process . For given positive constants , define the set of th conjunctions with the th largest order statistics of . In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions is not empty. Imposing the Albin's conditions on , in this paper we obtain an exact asymptotic expansion of this probability as tends to infinity. Furthermore, we establish the tail asymptotics of the supremum of the order statistics processes of skew-Gaussian processes and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes.