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Common Randomness Generation from Finite Compound Sources

Rami Ezzine, Moritz Wiese, Christian Deppe, Holger Boche

TL;DR

This paper addresses generating common randomness from finite compound sources with one-way, rate-limited communication. It introduces a single-letter framework yielding a lower bound and an upper bound on the compound CR capacity $C_{CCR}(R)$, governed by auxiliary variables $U$ that satisfy Markov chains with the source observations and a constraint $I(U;X)- min_s I(U;Y_s) \\le R$. The bounds collapse to $H(X)$ when $R$ is large enough (specifically $R \\ge \max_s H(X|Y_s)$), and two corollaries show additional scenarios where the bounds are tight. The results extend classical two-source CR capacity concepts to compound-source uncertainty, providing fundamental limits for CR generation under source-state ambiguity in one-way communication settings.

Abstract

We investigate the problem of generating common randomness (CR) from finite compound sources aided by unidirectional communication over rate-limited perfect channels. The two communicating parties, often referred to as terminals, observe independent and identically distributed (i.i.d.) samples of a finite compound source and aim to agree on a common random variable with a high probability for every possible realization of the source state. Both parties know the set of source states as well as their statistics. However, they are unaware of the actual realization of the source state. We establish a single-letter lower and upper bound on the compound CR capacity for the specified model. Furthermore, we present two special scenarios where the established bounds coincide.

Common Randomness Generation from Finite Compound Sources

TL;DR

This paper addresses generating common randomness from finite compound sources with one-way, rate-limited communication. It introduces a single-letter framework yielding a lower bound and an upper bound on the compound CR capacity , governed by auxiliary variables that satisfy Markov chains with the source observations and a constraint . The bounds collapse to when is large enough (specifically ), and two corollaries show additional scenarios where the bounds are tight. The results extend classical two-source CR capacity concepts to compound-source uncertainty, providing fundamental limits for CR generation under source-state ambiguity in one-way communication settings.

Abstract

We investigate the problem of generating common randomness (CR) from finite compound sources aided by unidirectional communication over rate-limited perfect channels. The two communicating parties, often referred to as terminals, observe independent and identically distributed (i.i.d.) samples of a finite compound source and aim to agree on a common random variable with a high probability for every possible realization of the source state. Both parties know the set of source states as well as their statistics. However, they are unaware of the actual realization of the source state. We establish a single-letter lower and upper bound on the compound CR capacity for the specified model. Furthermore, we present two special scenarios where the established bounds coincide.
Paper Structure (10 sections, 4 theorems, 76 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. System Model, Definitions and Main Result
  3. System Model
  4. Achievable Rate and Capacity
  5. Main Result
  6. Proof of the Lower-bound in Theorem \ref{['compound CRcapacity']}
  7. Proof of the upper-bound in Theorem \ref{['compound CRcapacity']}
  8. Proof of Corollary \ref{['corollary1']}
  9. Proof of Corollary \ref{['corollary2']}
  10. Conclusion

Key Result

Theorem 1

For the model in Fig CRprotocol, the compound CR capacity $C_{CCR}(R)$ satisfies and

Figures (1)

  • Figure 1: Two-source model for CR generation from finite compound sources with one-way communication over rate-limited perfect channels.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Lemma 1