High level exceedances in stationary sequences with extremal index (Q1110897)

Let \(\{\xi_ n\}\) be a stationary sequence satisfying an analogue of Leadbetter's \(D(u_ n)\) condition and possessing extremal index \(\theta\), \(0<\theta \leq 1\). Denote by \(S_ n\) the number of exceedances of a level \(u_ n\) by \(\xi_ 1,...,\xi_ n\), i.e., the number of i, \(1\leq i\leq n\), such that \(\xi_ i>u_ n.\) The author proves that the limit laws of \(S_ n\) is of the compound Poisson type and specifies the parameters. She also determines the joint limit laws for exceedances of multiple levels, and consequently, for finite numbers of upper order statistics.