Large Deviations for Products of Empirical Measures of Dependent Sequences

We prove large deviation principles (LDP) for m-fold products of empirical measures and for U-empirical measures, where the underlying sequence of random variables is a special Markov chain, an exchangeable sequence, a mixing sequence or an independent, but not identically distributed, sequence. The LDP can be formulated on a subset of all probability measures, endowed with a topology which is even finer than the usual τ -topology. The advantage of this topology is that the map ν → ∫ Sm φdν is continuous even for certain unbounded φ taking values in a Banach space. As a particular application we get large deviation results for U-statistics and V -statistics based on dependent sequences. Furthermore, we prove an LDP for products of empirical processes in a topology, which is finer than the projective limit τ -topology.