Tight Weighted Second-Order Asymptotics for the Wyner--Ahlswede--Körner Problem Under Regular Posterior Geometry

Abstract

This paper determines the exact weighted normal approximation for the finite-alphabet Wyner--Ahlswede--Körner problem under local regularity of the posterior optimization. Liu's type-based achievability is governed by the variance of the weighted optimizer information density, whereas the known converse dispersion bound retains only the variance of its conditional expectation given the source pair. We show that the missing conditional-variance term is a genuine fixed-composition fluctuation. The converse first represents the auxiliary-variable optimization as a convexification problem on the posterior simplex and uses the associated dual deficit to quantify the suboptimality of code-induced posteriors. After conditioning on a joint type, a random deletion process yields an exact likelihood decomposition into a support-function score, a nonnegative predictable deficit, and a martingale. Posterior localization and barycentric inversion identify the martingale's predictable variance, while a variance-completion construction permits a martingale central limit theorem without conditioning on a terminal event. Averaging the fixed-type Gaussian bound over empirical joint types gives a total dispersion equal to the achievability variance. The uniqueness requirement is further relaxed to a variance-identifiability condition over all optimal posterior decompositions. A binary symmetric specialization verifies the assumptions and gives a closed form.