# zbMATH — the first resource for mathematics

Let $$\alpha_ n(s)=n^{1/2}(F_ n(s)-s)$$ (0$$\leq s\leq 1$$, $$n=1,2,...)$$ resp. $$\beta_ n(s)$$ be the uniform empirical resp. quantile process. Further let $$R_ n(s)=\alpha_ n(s)+\beta_ n(s)$$ and $$b_ n=n^{1/4}(\log n)^{-1/2}$$. J. Kiefer [Deviations between the sample quantile process and the sample d.f., in: Proc. Symp., Indiana Univ., Bloomington, Ind. (1969), 299-319 (1970)] claimed that $b_ n\sup | R_ n(s)| /(\sup | \beta_ n(s)|)^{1/2}\to 1\quad a.s.$ The authors give a proof of this statement. They present an analogous result replacing $$\alpha_ n$$ resp. $$\beta_ n$$ by a standardized partial sum resp. renewal process. Weighted versions of the above results are also presented.