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