Convergence of the Point Process of Exceedances, and the Distribution of kth Largest Maxima

Abstract

In this chapter we return to the general situation and notation of Chapter 3 and consider the points; (regarded as “time instants”) at which the general stationary sequence {ξ_j} exceeds some given level u. These times of exceed-ance are stochastic in nature and may be viewed as a point process. Since exceedances of very high levels will be rare, one may suspect that this point process will take on a Poisson character at such levels. An explicit theorem along these lines will be proved and the asymptotic distributions of kth largest values (order statistics) obtained as corollaries. Generalizations of this theorem yield further results concerning joint distributions of kth largest values. The formal definition and simple properties of point processes which will be needed are given in the appendix.