Are cross-validated prediction errors i.i.d? Say, we test an arbitrary regression or classification procedure on $n$ independent samples with leave-one-out cross-validation. This results in an estimate of the prediction error $e_n$ for each sample $n$.
Can these $e_n$ be assumed to be independent draws of a (probably unknown) distribution?
My intuition says no, because (1) the training set is almost the same for each test sample, and (2) samples are used for both, training and testing.
If my intuition is wrong, and errors are independent, what about k-fold cross-validation, where the same training set is used for groups of $n/k$ samples?

Disclaimer: I tried to ask this question as concisely and generally as possible. If it lacks detail or specifity, please comment and I  will update the question accordingly.
 A: They can't be independent. Consider adding one extreme outlier sample, then many of your cross validation folds will be skewed in a correlated way.
A: I think you need to be clear what distribution you need to represent. This differers according to what the cross validation is meant for. 


*

*In the case that the cross validation is meant to  measure (approximate) the performance of the model obtained from this particular training set, the corresponding distribution would be the distribution of cases in the training set at hand. From that perspective, you draw almost the entire population, though without replacement.    

*In contrast, if you are asking about the distribution of $n$ cases drawn from the population the training set was drawn from, then the cross validation resampled surrogate training sets are correlated. See e.g. Bengio, Y. and Grandvalet, Y.: No Unbiased Estimator of the Variance of K-Fold Cross-Validation Journal of Machine Learning Research, 2004, 5, 1089-1105.
This is important for comparisons which algorithm performs better for a particular type of data.
