EXTREMES OF ORDER STATISTICS OF STATIONARY GAUSSIAN PROCESSES

Let {X-i(t); t >= 0}, 1 <= i <= n, be mutually independent and identically distributed centered stationary Gaussian processes. Under some mild assumptions on the covariance function, we derive an asymptotic expansion of P (sup(t subset of[0,xmr(u)]) X-(r)(t) <= u) as u -> infinity, where m(r)(u) = (P(sup(t is an element of[0,1]) X-(r)(t) > u))(-1) (1 + o(1)), and {X-(r)(t); t >= 0} is the rth order statistic process of {X-i(t); t >= 0}, 1 <= i; r <= n. As an application of the derived result, we analyze the asymptotics of supremum of the order statistic process of stationary Gaussian processes over random intervals.