On the asymptotic locations of the largest and smallest extremes of a stationary sequence
This article deals with the asymptotic joint locations of the \(k\)-th largest and the \(r\)-th smallest order statistics of a stationary sequence of random variables. It is shown under some conditions, that the upper and lower order statistics are asymptotically independent, by showing that the corresponding limiting point processes are couplings of independent compound Poisson point processes. The asymptotic joint distribution of the locations of the \(k\)-th largest and \(r\)-th smallest extremes are obtained and it is shown that, under some conditions, the locations vector of the \(k\)-th upper order statistics and the locations vector of the \(r\)-th lower order statistics are asymptotically independent. The results are extensions of the results in [\textit{L. Pereira} and \textit{H. Ferreira}, J. Stat. Plann. Inference 104, No. 2, 287--295 (2002; Zbl 0998.60047)], for the sequences of partial maximum and minimum and their locations.
scientific article; zbMATH DE number 6919751