Exceedances of Levels and kth Largest Maxima

In this chapter, we investigate properties of the exceedances of levels {u _n } by ξ_i, ξ₂,…, i.e. the points i for which ξ_i, > u_n, and as consequences, obtain limiting distributional results for the kth largest value among ξ₁,…, ξ₁. In particular, when the familiar assumption $$n\left( {1 - F\left( {u_n } \right)} \right) \to \tau \left( {0 < \tau < \infty } \right)$$ holds (Equation (1.5.1)), it will be shown that the exceedances take on a Poisson character as n becomes large. This leads to the limiting distributions for the kth largest values for any fixed rank k = 1, 2,… (the kth “extreme order statistics”) and to their limiting joint distributions.