  A collection of quantile curves provides a complete picture of conditional
distributions. Properly centered and scaled versions of estimated curves at
various quantile levels give rise to the so-called quantile regression process
(QRP). In this paper, we establish weak convergence of QRP in a general series
approximation framework, which includes linear models with increasing
dimension, nonparametric models and partial linear models. An interesting
consequence is obtained in the last class of models, where parametric and
non-parametric estimators are shown to be asymptotically independent.
Applications of our general process convergence results include the
construction of non-crossing quantile curves and the estimation of conditional
distribution functions. As a result of independent interest, we obtain a series
of Bahadur representations with exponential bounds for tail probabilities of
all remainder terms. Bounds of this kind are potentially useful in analyzing
statistical inference procedures under divide-and-conquer setup.
