The joint distribution of a random vector (X, Y) in ^p is usually described through the conditional quantiles of Y given X = x. In this article, we rather concentrate on the quantiles of Y conditioned by X ≤ x. Such quantiles can be estimated simply by inverting the empirical version of the cumulative distribution function (cdf) of Y given X ≤ x. In Aragon et al.. [Aragon, Y., Daouia, A. and Thomas-Agnan, C. (2002). Nonparametric frontier estimation: A conditional quantile-based approach. Discussion paper, GREMAQ et LSP, Universit de Toulouse (http://www.univ-tlse1.fr/GREMAQ/Statistique/adt1202.pdf). Forthcoming in Econometric Theory.], the study of these empirical conditional quantiles was initiated in the context of estimating the production frontier, i.e., the set of the most efficient firms in a production technology. The weak consistency and asymptotic normality have been proved. This article is mainly devoted to prove an asymptotic linear representation, in the almost sure sense, for these nonparametric quantiles in terms of the conditional empirical cdf, and to provide an asymptotic bound for the error term involved. A Donsker-type functional convergence result is proved for the conditional empirical cdf and consequently, a theorem on the asymptotic limit of the conditional empirical quantile process is derived.