Extremes of order statistics of stationary processes

Let (Formula presented.) be independent copies of a stationary process (Formula presented.). For given positive constants u,T, define the set of rth conjunctions (Formula presented.) with (Formula presented.) the rth largest order statistics of (Formula presented.). In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions (Formula presented.) is not empty. Imposing the Albin’s conditions on X, in this paper we obtain an exact asymptotic expansion of this probability as u 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.