### Abstract

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.

Original language | English (US) |
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Article number | 6665138 |

Pages (from-to) | 881-903 |

Number of pages | 23 |

Journal | IEEE Transactions on Information Theory |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2014 |

### Keywords

- Dispersion
- Slepian-Wolf
- asymmetric broadcast channel
- multiple-access channel
- network information theory
- second-order coding rates

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

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## Cite this

*IEEE Transactions on Information Theory*,

*60*(2), 881-903. [6665138]. https://doi.org/10.1109/TIT.2013.2291231