On the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics (Q1094021)

We shall disclose a relationship between the almost sure stability of weighted empirical distribution functions and sums of order statistics. First we obtain an extension of a theorem due to \textit{E. Csáki} [Nonparametric statistical inference, Budapest 1980, Vol. I, Colloq. Math. Soc. János Bolyai 32, 123-138 (1982; Zbl 0511.62049)] on the almost sure stability of the standardized uniform empirical distribution function. This result is then shown to be an essential tool to derive a characterization of the almost sure stability of the sum of \(k_ n\) upper order statistics from a sample of n independent observations from a distribution with positive support in the domain of attraction of a non- normal stable law, where \(1\leq k_ n\leq n\) and \(k_ n\to \infty\) as \(n\to \infty\).