On the probability of conjunctions of stationary Gaussian processes

Let {Xi(t),t≥0},1≤i≤n be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants u,T, define the set of conjunctions C[0,T],u≔{t∈[0,T]:min1≤i≤nXi(t)≥u}. Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of P{C[0,T],u≠ϕ}, as u→∞. Moreover, we establish the Berman sojourn limit theorem for the random process {min1≤i≤nXi(t),t≥0} and derive the tail asymptotics of the supremum of each order statistics process.