Invariance principles for sums of extreme sequential order statistics attracted to Lévy processes
Let \(X_{n,i}\), \(1\leq i \leq n\), be a rowwise i.i.d. triangular array of random variables and let \(X_{i:[nt]}^{(n)}\) denote the \(i\)th smallest sequential order statistic of \(X_{n,1}, \dots , X_{n,[nt]}, \;0\leq t \leq 1.\) Strong convergence results and the invariance principle for centered partial sums of the first \(k\) sequential order statistics \(X_{i:[nt]}^{(n)}\) attracted to an infinitely divisible law are established. The quantile approach based on a representation of order statistics of i.i.d. uniformly distributed random variables is used to prove the joint strong convergence of sequential order statistics to extremal processes.
scientific article; zbMATH DE number 1792503
scientific article