Clustering of upcrossings of high values

We determine the class of point processes which may occur as limits in distribution of the T-upcrossings point process N n ( T ) ( B ) = ∑ i = 1 n 1 { X i ≤ u n < X i + T } δ i / n ( B ) , n ≥ 1 , B ⊂ [ 0 , 1 ] , of real levels u = { u n } n ≥ 1 by the random variables of a stationary sequence X = { X n } n ≥ 1 . tensity of the point process limit of N n ( T ) is linked to a parameter η ( T ) ∈ [ 0 , 1 ] here called the T-upcrossings index. The set of indices { η ( T ) , T ≥ 1 } are measures of clustering of T-upcrossings of u by the variables in X and, as we prove, are related to each other and with the extremal index. This T-upcrossings index is of great importance when we are interested in the mean number of upcrossings, that occur in particular instants, of high thresholds. We give explicit formulas for the T-upcrossings index under local dependence conditions. Furthermore, a detailed analysis of the effect that systematic sub-sampling has on the value of this index is presented. sults are illustrated with two autoregressive processes.