If a set of models is being compared using BIC and AIC, given the fact that the true model (the one which generated the data) is in this set (and given the other assumptions that guarantee BIC consistency in selecting the true model);
As N goes to infinity, if I understand things correctly, BIC will always pick the true model, AIC will either pick the true model or a slightly more general one with more parameters. This would mean that BIC will always pick a model that predics as well or better that the one picked by AIC.
Q1) So, can we say that as N goes to infinity, BIC will perfom better that AIC not only form an "explanation" point of view (selecting the correct model) but also from a "prediction" point of view (selecting the model that minimizes the prediction error)?
Q2) I know there are some situations where the true model is not the best for prediction, but when this happens the best should have less parameters and not more. Anyway, will this situasions happen only if N is small or are there examples of this happening as N goes to infinity?