Weak convergence of the tail empirical process for dependent sequences (Q1004402)

Let \(\xi_1,\ldots, \xi_n\) be a stationary sequence of random observations fulfilling some mixing condition. For a given threshold \(u_n\) (becoming large) and a scaling factor \(\sigma_n > 0\), the tail empirical distribution function is defined as \(\tilde T_n(x) = \frac{1}{n\bar{F}(u_n)} \sum_{i=1}^n 1_{\{ \xi_i > u_n + x\sigma_n\}}\). Here \(F\) is the marginal distribution of the \(\xi\)'s and \(\bar{F} = 1 - F\) is the associated tail distribution function. The paper provides conditions under which \(\tilde T_n\) properly standardized converges to a Gaussian limit in the Skorokhod space \(D\). An interesting modification considers the situation when \(u_n\) is replaced by appropriate order statistics of the sample.