Renyi entropy, guesswork moments, and large deviations

For a large class of stationary probability measures on A^N, where A is a finite alphabet, we compute the specific Renyi entropy of order α and the specific guesswork moments of order β > -1. We show that the specific guesswork moment of order β equals the specific Renyi entropy of order α = 1 / (1 + β) multiplied by β. The method is based on energy-entropy estimates suggested by statistical physics. The technique also yields a simple proof of the large deviation principle for the empirical measure on the space of an irreducible sofic shift with reference probability measure ν, which is stationary and satisfies a rate condition on the probability of allowed words.