Dominating Points and Entropic Projections

Some Conditional Laws of Large Numbers (CLLN) are related to minimization problems: the limit of the CLLN is the minimizer of a large deviation rate function on the limiting conditioning set. When the CLLN is concerned with empirical means, the minimizer is called a dominating point; if it is concerned with empirical measures, it is called an entropic projection. CLLNs are obtained both for empirical means and measures with independent random weights. By means of convex conjugate duality, one obtains dual equalities and dual representations of the minimizers: the dominating points and the entropic projections. For some convex conditioning events, it may happen that usual integral representations of the dominating point fail: no entropic projection exists. This phenomenon is clarified by introducing extended minimization problems the minimizers of which may not be measures anymore. It appears that in some situations, the generalized entropic projections discovered by Csiszár are the measure component of these extended minimizers. The important case of relative entropy is studied in details.