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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 38, Fasc. 4,
pages 61 - 75
DOI: 10.19195/0208-4147.38.1.4
 

EXTREMES OF ORDER STATISTICS OF STATIONARY GAUSSIAN PROCESSES

Chunming Zhao

Abstract: 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( sup X (r)(t) ≤ u) asu → ∞, t∈[0,xmr(u)]
where
-1 mr (u) = (ℙ( st∈u[p0,1]X (r)(t) > u)) (1+ o(1)),
and (X (r)(t),t ≥ 0) is the r th 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.

2010 AMS Mathematics Subject Classification: Primary: 60G15; Secondary: 60G70.

Keywords and phrases: Asymptotic, Gaussian processes, order statistic, stationarity, supremum.

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