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In model selection, several types of cross-validation are commonly used andmany variants have been introduced. While consistency of some of these methodshas been proven, their rate of convergence to the oracle is generally stillunknown. Until now, an asymptotic analysis of crossvalidation able to answerthis question has been lacking. Existing results focus on the ''pointwise''estimation of the risk of a single estimator, whereas analysing model selectionrequires understanding how the CV risk varies with the model. In this article,we investigate the asymptotics of the CV risk in the neighbourhood of theoptimal model, for trigonometric series estimators in density estimation.Asymptotically, simple validation and ''incomplete'' V --fold CV behave likethe sum of a convex function fn and a symmetrized Brownian changed in time Wgn/V. We argue that this is the right asymptotic framework for studying modelselection.

Local asymptotics of cross-validation in least-squares density estimation