The paper investigates the problem of common randomness (CR) generation in a two-party communication setting, where a sender (Alice) and a receiver (Bob) 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.
The key contributions are:
The authors 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.
The proof utilizes a generalized typicality concept suitable for Polish alphabets, which was introduced in prior work. This typicality generalizes the notion of strong typicality used for finite alphabets.
The transition to infinite Polish alphabets has significant consequences in terms of Shannon entropy convergence, variational distance convergence, and potential discontinuities in the capacity characterization, which are discussed.
For the case when the joint probability distribution is discrete and the alphabets are finite, the bounds coincide as studied in prior work.
The discontinuity behavior observed in the capacity characterization is a common phenomenon in information theory for increasingly complex communication scenarios involving arbitrary distributions on infinite alphabets. Determining the exact points of discontinuity and the corresponding capacity values remains an open problem.
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by Wafa Labidi,... at arxiv.org 05-06-2024
https://arxiv.org/pdf/2405.01948.pdfDeeper Inquiries