On the Probability of Conjunctions of Stationary Gaussian Processes

Abstract: Let ${X_i(t),t\ge0}, 1\le i\le n$ 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 $C_{[0,T],u}:={t\in [0,T]: \min_{1 \le i \le n} X_i(t) \ge u}.$ Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of $ P(C_{[0,T],u} \not=\varphi)$ as $u\to\infty$. Moreover, we establish the Berman sojourn limit theorem for the random process ${\min_{1 \le i \le n} X_i(t), t\ge0}$ and derive the tail asymptotics of the supremum of each order statistics process.