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Bahadur-Kiefer-type processes. (English) Zbl 0712.60028
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.
Reviewer: P.Révész

MSC:
60F15 Strong limit theorems
60F05 Central limit and other weak theorems
62G30 Order statistics; empirical distribution functions
60F17 Functional limit theorems; invariance principles
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