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On the Dispersions of Three Network Information Theory Problems

机译：关于三个网络信息理论问题的分散性

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摘要

We analyze the dispersions of distributed lossless source coding (the Slepian–Wolf problem), the multiple-access channel, and the asymmetric broadcast channel. For the two-encoder Slepian–Wolf problem, we introduce a quantity known as the entropy dispersion matrix, which is analogous to the scalar dispersions that have gained interest recently. We prove a global dispersion result that can be expressed in terms of this entropy dispersion matrix and provides intuition on the approximate rate losses at a given blocklength and error probability. To gain better intuition about the rate at which the nonasymptotic rate region converges to the Slepian–Wolf boundary, we define and characterize two operational dispersions: 1) the local dispersion and 2) the weighted sum-rate dispersion. The former represents the rate of convergence to a point on the Slepian–Wolf boundary, whereas the latter represents the fastest rate for which a weighted sum of the two rates converges to its asymptotic fundamental limit. Interestingly, when we approach either of the two corner points, the local dispersion is characterized not by a univariate Gaussian, but a bivariate one as well as a subset of off-diagonal elements of the aforementioned entropy dispersion matrix. Finally, we demonstrate the versatility of our achievability proof technique by providing inner bounds for the multiple-access channel and the asymmetric broadcast channel in terms of dispersion matrices. All our proofs are unified by a so-called vector rate redundancy theorem, which is proved using the multidimensional Berry–Esséen theorem.
机译：我们分析了分布式无损源编码（Slepian–Wolf问题），多路访问信道和非对称广播信道的色散。对于两个编码器的Slepian-Wolf问题，我们引入了一个称为熵色散矩阵的量，该量类似于最近引起人们关注的标量色散。我们证明了可以用该熵色散矩阵表示的全局色散结果，并提供了在给定块长和错误概率下近似速率损失的直觉。为了更好地了解非渐近速率区域收敛到Slepian-Wolf边界的速率，我们定义并描述了两个操作色散：1）局部色散和2）加权总和速率色散。前者表示收敛到Slepian–Wolf边界上某一点的速率，而后者则表示最快的速率，这两种速率的加权总和收敛到其渐近基本极限。有趣的是，当我们接近两个角点中的任意一个时，局部色散的特征不是单变量的高斯分布，而是二元变量以及上述熵色散矩阵的非对角元素的子集。最后，我们通过在色散矩阵方面为多路访问信道和非对称广播信道提供内边界来证明我们可实现性证明技术的多功能性。我们所有的证明都是通过所谓的矢量率冗余定理统一起来的，该定理是使用多维Berry-Esséen定理证明的。

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