### 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) |
---|---|

Article number | 6665138 |

Pages (from-to) | 888-903 |

Number of pages | 16 |

Journal | IEEE Transactions on Information Theory |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*IEEE Transactions on Information Theory*,

*60*(2), 888-903. [6665138]. https://doi.org/10.1109/TIT.2013.2291231

**On the dispersions of three network information theory problems.** / Tan, Vincent Y F; Kosut, Oliver.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 60, no. 2, 6665138, pp. 888-903. https://doi.org/10.1109/TIT.2013.2291231

}

TY - JOUR

T1 - On the dispersions of three network information theory problems

AU - Tan, Vincent Y F

AU - Kosut, Oliver

PY - 2014/2

Y1 - 2014/2

N2 - 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.

AB - 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.

KW - asymmetric broadcast channel

KW - Dispersion

KW - multiple-access channel

KW - network information theory

KW - second-order coding rates

KW - Slepian-Wolf

UR - http://www.scopus.com/inward/record.url?scp=84893459202&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893459202&partnerID=8YFLogxK

U2 - 10.1109/TIT.2013.2291231

DO - 10.1109/TIT.2013.2291231

M3 - Article

AN - SCOPUS:84893459202

VL - 60

SP - 888

EP - 903

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 2

M1 - 6665138

ER -