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We prove the large deviation principle for several entropy and cross entropyestimators based on return times and waiting times on shift spaces over finitealphabets. In the case of standard return times, we obtain a nonconvexlarge-deviation rate function. We consider shift-invariant probability measuressatisfying some decoupling conditions which imply no form of mixing norergodicity. We establish precise relations between the rate functions of thedifferent estimators, and between these rate functions and the correspondingpressures, one of which is the R\'enyi entropy function. The results apply inparticular to irreducible Markov chains, equilibrium measures for Bowen-regularpotentials, g-measures, invariant Gibbs states for summable interactions instatistical mechanics, and also to probability measures that may be far fromGibbsian, including some hidden Markov models and repeated quantum measurementprocesses.

Large deviations of return times and related entropy estimators on shift spaces