Common Randomness Generation From Finite Compound Sources Aided by One-Way Communication

Abstract

We investigate the problem of generating common randomness (CR) from a finite compound source aided by unidirectional communication over a rate-limited perfect channel. The two communicating parties observe independent and identically distributed (i.i.d.) samples of a finite compound source and aim to agree on a common random variable with high probability for every possible state. Both parties know the set of source states as well as their statistics. However, they don’t know the actual state. We establish a single-letter formula for the compound CR capacity in the presence of communication over the channel and study key properties of the compound CR capacity: super-additivity, concavity, and continuity. We also consider the case where there is no communication between the terminals, and only the source outputs observed by the terminal at the receiving end of the perfect channel are state-dependent. In this setting, we establish single-letter bounds on the compound CR capacity. The single-letter lower bound is derived under the assumption that the source distributions are pairwise distinct for all states. Finally, within the same setting, we propose a CR generation scheme for a two-state binary source example. Notably, this scheme does not depend on the previously mentioned assumption.