A note on exceedances and rare events of non-stationary sequences

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

A note on exceedances and rare events of non-stationary sequences

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.