Bahadur representation for U-quantiles of dependent data

U-quantiles are applied in robust statistics, like the Hodges-Lehmann estimator of location for example. They have been analysed in the case of independent random variables with the help of a generalized Bahadur representation. Our main aim is to extend these results to U-quantiles of strongly mixing random variables and functionals of absolutely regular sequences. We obtain the central limit theorem and the law of the iterated logarithm for U-quantiles as straightforward corollaries. Furthermore, we improve the existing result for sample quantiles of mixing data.