On the Dispersions of Three Network Information Theory Problems

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.