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Extremes of order statistics of stationary Gaussian processes

Zhao C.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

61--75

Let {Xi(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 [formula] ]X(r) (t) ≤ u) as u → ∞, where mr(u) = (P([formula] X(r) (t) > u))−1 (1 + o(1)), and {X(r) (t), t ≥ 0} is the rth order statistic process of {Xi(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.