On the rate of convergence of the sum of the sample extremes (Q1111269)

Let \(X_ 1,X_ 2,..\). be independent random variables with a common distribution function in the domain of normal attraction of the completely asymmetric stable law F(\(\cdot;\alpha,1)\) with \(0<\alpha <1\). Let \(x_{n:1}\leq X_{n:2}\leq...\leq X_{n:n}\) denote the order statistics of \(X_ 1,X_ 2,...,X_ n\). The author derives the rate of convergence of \[ n^{-1/\alpha}(X_{n:n}+...+X_{n:n-k_ n+1}) \] to F(\(\cdot;\alpha,1)\) as both n and \(k_ n\to \infty\). As a consequence he obtains a representation of the sum \(X_{n:n}+...+X_{n:n-k_ n+1}\).