正则后验几何下Wyner-Ahlswede-Körner问题的紧加权二阶渐近分析
Exact Second-Order Asymptotics for the Wyner--Ahlswede--Körner Problem
AI总结：
本文针对正则后验几何下的Wyner-Ahlswede-Körner问题，确定其紧加权二阶渐近的精确加权正态近似，补充缺失的条件方差项，将色散界与可达性方差统一，二元对称特例验证假设并给出闭式。
AI中文摘要：
本文确定了有限字母集Wyner-Ahlswede-Körner问题在后验优化局部正则性下的精确加权正态近似。Liu的基于类型的可达性由加权优化器信息密度的方差决定，而已知的 converse 色散界仅保留其在源对条件下的条件期望方差。我们证明缺失的条件方差项是真实的固定组成波动。Converse首先将辅助变量优化表示为后验单纯形上的凸化问题，并使用相关对偶赤字量化码诱导后验的次优性。在联合类型条件下，随机删除过程将精确似然分解为支撑函数得分、非负可预测赤字和鞅。后验定位和重心反演确定鞅的可预测方差，而方差补全构造允许不依赖终端事件条件的鞅中心极限定理。将固定类型高斯界对经验联合类型求平均，得到总色散等于可达性方差。唯一性要求进一步放宽为所有最优后验分解上的方差可识别性条件。二元对称特例验证了假设并给出闭式表达。
英文摘要：
We determine the exact local second-order rate region of the finite-alphabet Wyner--Ahlswede--Körner 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.
