On the extreme order statistics for a stationary sequence

AbstractSuppose that {ξj} is a strictly stationary sequence which satisfies the strong mixing condition. Denote by M(k)n the kth largest value of ξ1,ξ2,…,ξn, and {υn(·)} a sequence of normalizing functions for which P[M(1)n⩽υn(x)]converges weakly to a continuous distribution G(x). It is shown that if for some k=2,3,…,P[M(k)n⩽υn(x)] converges for each x, then there exist probabilities p1,…,pk−1 such that P[M(j)n⩽υn(x)] converges weakly to G(x)1+∑i=1j−i(-logG(x))ii!Pi for j=2,…,k, where natural interpretations can be given for the pj. This generalizes certain results due to Dziubdziela (1984) and Hsing, Hüsler and Leadbetter (1986). It is further demonstrated that, with minor modification, the technique can be extended to study the joint limiting distribution of the order statistics. In particular, Theorem 1 of Welsch (1972) is generalized, and some links between the convergence of the order statistics and that of certain point processes are established