Extremes of order statistics of stationary processes

Let $$\{X_i(t),t\ge 0\}, 1\le i\le n$$ { X i ( t ) , t ≥ 0 } , 1 ≤ i ≤ n be independent copies of a stationary process $$\{X(t), t\ge 0\}$$ { X ( t ) , t ≥ 0 } . For given positive constants $$u,T$$ u , T , define the set of $$r$$ r th conjunctions $$ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$$ C r , T , u : = { t ∈ [ 0 , T ] : X r : n ( t ) > u } with $$X_{r:n}(t)$$ X r : n ( t ) the $$r$$ r th largest order statistics of $$X_i(t), t\ge 0, 1\le i\le n$$ X i ( t ) , t ≥ 0 , 1 ≤ i ≤ n . In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $$C_{r,T,u}$$ C r , T , u is not empty. Imposing the Albin’s conditions on $$X$$ X , in this paper we obtain an exact asymptotic expansion of this probability as $$u$$ 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. Copyright Sociedad de Estadística e Investigación Operativa 2015