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A collection of quantile curves provides a complete picture of conditionaldistributions. Properly centered and scaled versions of estimated curves atvarious quantile levels give rise to the so-called quantile regression process(QRP). In this paper, we establish weak convergence of QRP in a general seriesapproximation framework, which includes linear models with increasingdimension, nonparametric models and partial linear models. An interestingconsequence is obtained in the last class of models, where parametric andnon-parametric estimators are shown to be asymptotically independent.Applications of our general process convergence results include theconstruction of non-crossing quantile curves and the estimation of conditionaldistribution functions. As a result of independent interest, we obtain a seriesof Bahadur representations with exponential bounds for tail probabilities ofall remainder terms. Bounds of this kind are potentially useful in analyzingstatistical inference procedures under divide-and-conquer setup.

Quantile Processes for Semi and Nonparametric Regression