Asymptotic \((r-1)\)-dependent representation for \(r\)th order statistic from a stationary sequence (Q1802317)

Let \(X_ 1,\dots,X_ n\) be a stationary sequence. Its \(k\)th largest value is denoted by \(M_ n^{(k)}\). The author derives conditions for the existence of an \((r-1)\)-dependent stationary sequence \(\tilde X_ 1,\dots,\tilde X_ n\) such that \(\sup_ x| P(M_ n^{(k)}\leq x)- P(\tilde M_ n^{(k)}\leq x)|\to 0\) as \(n\to\infty\) for \(0\leq k\leq r\) where \(\tilde M_ n^{(k)}\) are order statistics of \(\tilde X_ 1,\dots,\tilde X_ n\). Convergence of all order statistics is also considered.