On the asymptotic joint distribution of an unbounded number of sample extremes (Q1092548)

Convergence of the sample maximum to a nondegenerate random variable, as the sample size \(n\to \infty\), implies the convergence in distribution of the k largest sample extremes to a k-dimensional random vector \(M_ k\), for all fixed k. If we let \(k=k(n)\to \infty\), k/n\(\to 0\), then a question arises in a natural way: how fast can k grow so that asymptotic probability statements are unaffected when sample extremes are replaced by \(M_ k\). We answer this question for two classes of events - the class of all Lebesgue sets in \(R^ k\) and the class of events of the form \(\{x\in R^ k\), \(\sum ^{k}_{1}x_ i\leq a\}\).