Exact Second-Order Asymptotics for the Wyner--Ahlswede--K\"orner Problem

Abstract

We determine the exact local second-order rate region of the finite-alphabet Wyner--Ahlswede--K\"orner problem under full support, at informative boundary points exposed by a finite supporting slope. The characterization permits nonunique optimizing test channels with unequal information variances and requires no neighborhood smoothness of the optimized first-order value. The optimal success function is a conditional Gaussian envelope averaged over the fluctuation of the helper's observed source type. For each type, the helper selects the minimum or maximum conditional variance according to the remaining description budget, while preserving both first-order rates. The converse follows from a uniform Gaussian bound for ordinary channel images. An exact posterior score decomposition separates a common source score, a nonnegative optimality gap, a negligible source residual on low-cost histories, and a conditionally independent channel fluctuation. Conditional-type covering with two endpoint optimizers and one common binning map attains the bound. Under a strict first-order benefit from the helper, a separate limiting-rate argument also gives the unrestricted weighted second-order optimum. A strictly positive finite source exhibits a strict gain over every fixed-optimizer Gaussian expression, and an exact binary image calculation illustrates the limiting coefficient.