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Common Randomness Generation from Sources with Infinite Polish Alphabet

Wafa Labidi, Rami Ezzine, Moritz Wiese, Christian Deppe, Holger Boche

TL;DR

This work extends common randomness generation to sources with infinite Polish alphabets under one-way noisy-channel communication, deriving single-letter lower and upper bounds on the CR capacity that are tight except at at most countably many points. It builds on generalized typicality for Polish spaces and Wyner-Ziv‑style coding to achieve rates governed by $I(U;X)$ under the constraint $I(U;X)-I(U;Y) \le C(W)$ (plus vanishing terms), with the channel capacity $C(W)$ playing a key role in the trade-off. A notable contribution is the result that, if $I(X;Y)=\infty$, the CR capacity can be infinite for certain continuous-source models, illustrating a stark contrast with finite-alphabet cases. The findings provide a rigorous framework for CR generation with continuous and infinite alphabets, informing practical correlated randomness protocols in advanced communications contexts.

Abstract

We investigate the problem of common randomness (CR) generation in the basic two-party communication setting in which a sender and a receiver aim to agree on a common random variable with high probability. The terminals observe independent and identically distributed (i.i.d.) samples of sources with an arbitrary distribution defined on a Polish alphabet and are allowed to communicate as little as possible over a noisy, memoryless channel. We establish single-letter upper and lower bounds on the CR capacity for the specified model. The derived bounds hold with equality except for at most countably many points where discontinuity issues might arise.

Common Randomness Generation from Sources with Infinite Polish Alphabet

TL;DR

This work extends common randomness generation to sources with infinite Polish alphabets under one-way noisy-channel communication, deriving single-letter lower and upper bounds on the CR capacity that are tight except at at most countably many points. It builds on generalized typicality for Polish spaces and Wyner-Ziv‑style coding to achieve rates governed by under the constraint (plus vanishing terms), with the channel capacity playing a key role in the trade-off. A notable contribution is the result that, if , the CR capacity can be infinite for certain continuous-source models, illustrating a stark contrast with finite-alphabet cases. The findings provide a rigorous framework for CR generation with continuous and infinite alphabets, informing practical correlated randomness protocols in advanced communications contexts.

Abstract

We investigate the problem of common randomness (CR) generation in the basic two-party communication setting in which a sender and a receiver aim to agree on a common random variable with high probability. The terminals observe independent and identically distributed (i.i.d.) samples of sources with an arbitrary distribution defined on a Polish alphabet and are allowed to communicate as little as possible over a noisy, memoryless channel. We establish single-letter upper and lower bounds on the CR capacity for the specified model. The derived bounds hold with equality except for at most countably many points where discontinuity issues might arise.
Paper Structure (12 sections, 5 theorems, 17 equations, 1 figure)

This paper contains 12 sections, 5 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polish Alphabets
  4. Typicality Criteria for Polish Alphabets
  5. System Models, Definitions and Main Results
  6. System Model and Definitions
  7. Main Results
  8. Proof of Theorem \ref{['Theorem:Source1']}
  9. Proof of the Lower-Bound in Theorem \ref{['Theorem:Source1']}
  10. Proof of the Upper-Bound in Theorem \ref{['Theorem:Source1']}
  11. Conclusions
  12. Acknowledgments

Key Result

Theorem 1

(Portemanteau) Let $\mathcal{X}$ be a Polish space with Borel $\sigma-$algebra $\mathcal{E}_\mathcal{X}$. A bounded sequence of probability measures $P_n, \quad n = 1 , 2 , \ldots$ on $(\mathcal{X},\mathcal{E}_\mathcal{X})$ is said to converge weakly to a probability measure $P$ if any of the follow

Figures (1)

  • Figure 1: Memoryless source model with one-way communication over a memoryless channel.

Theorems & Definitions (12)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5: Mitran
  • Theorem 6: Mitran
  • Theorem 7: Mitran
  • Definition 8
  • Definition 9
  • Theorem 10
  • ...and 2 more