Asymptotic Expansions of Smooth R\'{e}nyi Entropies and Their Applications

This study considers the unconditional smooth R'{e}nyi entropy, the smooth conditional R'{e}nyi entropy proposed by Kuzuoka [\emph{IEEE Trans.\ Inf.
Theory}, vol.~66, no.~3, pp.~1674–1690, 2020], and a new quantity which we term the conditional smooth R'{e}nyi entropy. In particular, we examine asymptotic expansions of these entropies when the underlying source with its side-information is stationary and memoryless. Using these smooth R'{e}nyi entropies, we establish one-shot coding theorems of several information-theoretic problems: Campbell’s source coding, guessing problems, and task encoding problems, all allowing errors. In each problem, we consider two error formalisms: the average and maximum error criteria, where the averaging and maximization are taken with respect to the side-information of the source. Applying our asymptotic expansions to the derived one-shot coding theorems, we derive various asymptotic fundamental limits for these problems when their error probabilities are allowed to be non-vanishing. We show that, in non-degenerate settings, the first-order fundamental limits differ under the average and maximum error criteria. This is in contrast to a different but related setting considered by the present authors (for variable-length conditional source coding allowing errors) in which the first-order terms are identical but the second-order terms are different under these criteria.