A Bahadur–Kiefer law for the Nadaraya empiric-quantile processes

We study the a.s. behavior of Nadaraya's empiric-quantile processes $\widehat{R}_{n}(\,\cdot\,)$. Proceeding by invariance we exploit stochastic properties of $\|\widehat{R}_{n}\|$ and show that a Bahadur–Kiefer strong law holds for these processes demonstrating robustness with respect to the class of perturbed kernel empirical d.f.'s. Also, in the process of obtaining our result, we derive a Strassen-type law of the iterated logarithm which extends a theorem of Finkelstein and is likely to be of independent interest. In addition, a brief profile of applications is included.