High level exceedances in stationary sequences with extremal index
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.
limit laws for exceedances of multiple levels
scientific article; zbMATH DE number 4074066