On the Second- and Third-Order Asymptotics of Smooth Rényi Entropy and Their Applications

This study examines asymptotic expansions of the unconditional and conditional smooth Rényi entropies for a memoryless source. Using these smooth Rényi entropies, we establish one-shot coding theorems of several information-theoretic problems: Campbell's source coding, guessing, and task encoding problems, all allowing errors. Applying our asymptotic expansions to the derived one-shot coding theorems, we provide various asymptotic fundamental limits of these problems in the regime of non-vanishing error probabilities.