  We prove the large deviation principle for several entropy and cross entropy
estimators based on return times and waiting times on shift spaces over finite
alphabets. We consider shift-invariant probability measures satisfying some
decoupling conditions which imply no form of mixing nor ergodicity. We
establish precise relations between the rate functions of the different
estimators, and between these rate functions and the corresponding pressures,
one of which is the R\'enyi entropy function. For the most commonly used
definition of return times, the large-deviation rate function is proved to be
nonconvex, except in marginal cases. The results apply in particular to
irreducible Markov chains, equilibrium measures for Bowen-regular potentials,
g-measures, invariant Gibbs states for absolutely summable interactions in
statistical mechanics, and also to probability measures which may be far from
Gibbsian, including some hidden Markov models and repeated quantum measurement
processes.
