On the Probability of Conjunctions of Stationary Gaussian Processes

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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]: ₁ ≤ ᵢ ≤ ₙ Xᵢ(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 {₁ ≤ ᵢ ≤ ₙ Xᵢ(t), t≥0} and derive the tail asymptotics of the supremum of each order statistics process.