Bahadur-Kiefer-type processes
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 \(b_ n=n^{1/4}(\log n)^{-1/2}\). \textit{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.
scientific article; zbMATH DE number 4172049