On the extreme order statistics for a stationary sequence
Let \(\{\xi_ j\}\) be a strictly stationary sequence of random variables satisfying the strong mixing condition. Denote by \(M_ n^{(k)}\) the k- th largest value of \(\xi_ 1,\xi_ 2,...,\xi_ n\). Suppose there exist (possibly non-linear) normalizing functions \(v_ n\), \(n\geq 1\), for which \(P[M_ n^{(1)}\leq v_ n(x)]\) converges weakly to a continuous distribution function G. The author shows that if for some \(k=2,3,...\), \(P[M_ n^{(k)}\leq v_ n(x)]\) converges for each x, then there exist probabilities \(p_ 1,p_ 2,..\). such that \(P[M_ n^{(j)}\leq v_ n(x)]\) converges weakly to \[ G(x)[1+\sum^{j-1}_{i=1}((-\log G(x))^ i/i!)p_ i] \] for \(j=2,3,...k\), where natural interpretations can be given for the \(p_ j\). He also studies the limit of \[ P[M_ n^{(1)}\leq v_ n(x),\quad M_ n^{(k)}\leq v_ n(y)] \] for any fixed k, and discusses the connection of the convergence of the order statistics and that of certain point processes.
scientific article; zbMATH DE number 4067956