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On the Bahadur - Kiefer Representation for Intermediate Sample Quantiles

We investigate a Bahadur-Kiefer type representation for the p-th empirical quantile corresponding to a sample of n i.i.d. random variables, when 0<p<1 is a sequence which, in particular, may tend to 0 or 1, i.e. we consider the case of intermediate sample quantiles. We obtain an &#39;in probability&#39; version of the Bahadur -- Kiefer type representation for a $\kn$-th order statistic when $r_n=\kn\wedge (n-k_n)\to \infty$ under some mild regularity conditions, and an &#39;almost sure&#39; version under additional assumption that $\log n/r_n\to 0$, $\nty$. A representation for the sum of order statistics laying between the population p-quantile and the corresponding empirical quantile is also established.