Semi-Finite Length Analysis for Information Theoretic Tasks
This work addresses a theoretical gap in information theory by providing more precise asymptotic bounds, though it is incremental as it builds on existing expansion methods.
The paper tackles the problem of asymptotic expansions for optimal values in information-theoretic tasks, which previously had non-vanishing errors, by deriving expansions up to constant order for upper and lower bounds, clarifying the ranges with asymptotically vanishing errors.
We focus on the optimal value for various information-theoretical tasks. There are several studies for the asymptotic expansion for these optimal values up to the order $\sqrt{n}$ or $\log n$. However, these expansions have errors of the order $o(\sqrt{n})$ or $o(\log n)$, which does not goes to zero asymptotically. To resolve this problem, we derive the asymptotic expansion up to the constant order for upper and lower bounds of these optimal values. While the expansions of upper and lower bonds do not match, they clarify the ranges of these optimal values, whose errors go to zero asymptotically.