Large deviations for empirical entropies of Gibbsian sources

The entropy of an ergodic finite-alphabet process can be computed from a single typical sample path x_1^n using the entropy of the k-block empirical probability and letting k grow with $n$ roughly like log n. We further assume that the distribution of the process is a g-measure; g-measures form a large class of Gibbs measures. We prove large deviation principles for conditional, non-conditional and relative k(n)-block empirical entropies.