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Strong laws for local quantile processes. (English) Zbl 0902.60027

Functional laws of the iterated logarithm for local quantile processes are established which are used to investigate the rate of almost sure convergence to zero of increments of size \(h_n\) of the uniform quantile process in the neighborhood of a fixed point \(t\in [0,1)\). In the range \(h_n\to 0\) and \(nh_n/\log n\to\infty\) as \(n\to\infty\) and for \(t\neq 0\) this rate is, somewhat unexpectedly, different from the corresponding rate for the uniform empirical process. As an application of his results the author shows that the best possible uniform almost sure rate of approximation of the uniform quantile process by a normalized Kiefer process is not better than \(O(n^{-1/4}(\log n)^{-\varepsilon})\), thus settling one of the open problems in the field of strong approximations.

MSC:

60F15 Strong limit theorems
60G15 Gaussian processes
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