Extremes of order statistics of stationary Gaussian processes

摘要

Let {Xi(t); t > 0}, 1 6 i 6 n, be mutually independentand identically distributed centered stationary Gaussian processes. Undersome mild assumptions on the covariance function, we derive an asymptoticexpansion ofP(supt2[0;xmr(u)]X(r)(t) 6 u)as u ! 1;wheremr(u) =(P( supt2[0;1]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 6 i; r 6 n. As an application of the derived result, we analyze the asymptoticsof supremum of the order statistic process of stationary Gaussian processesover random intervals.