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Capacity Bounds for Common Randomness Generation from Sources with Infinite Polish Alphabet over Noisy Memoryless Channels


Core Concepts
The paper establishes single-letter lower and upper bounds on the common randomness (CR) capacity for a two-source model involving correlated arbitrary sources with infinite Polish alphabets, assisted by one-way communication over noisy memoryless channels.
Abstract
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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Deeper Inquiries

What are the potential applications of the derived common randomness generation capacity bounds in practical communication systems

The derived common randomness generation capacity bounds can have significant implications for practical communication systems in various ways. One potential application is in the design and optimization of secure communication protocols that rely on common randomness for key generation. By understanding the limits of CR capacity, system designers can develop more efficient and reliable cryptographic schemes that leverage shared random information between communicating parties. This can enhance the security and privacy of data transmission in applications such as secure messaging, financial transactions, and data storage. Furthermore, the CR capacity bounds can also be utilized in the development of robust and efficient identification schemes for communication networks. By leveraging the insights from the derived bounds, researchers and engineers can enhance the performance of identification protocols, leading to faster and more reliable authentication processes in networked systems. This can be particularly valuable in scenarios where quick and secure identification is crucial, such as in IoT devices, access control systems, and secure data sharing platforms. Moreover, the understanding of CR capacity bounds can aid in optimizing resource allocation in communication systems. By knowing the limits of common randomness generation under specific constraints, system operators can allocate resources more effectively to achieve higher levels of reliability and efficiency in data transmission. This can lead to improved network performance, reduced latency, and enhanced overall system resilience in dynamic and challenging communication environments.

How can the discontinuity behavior observed in the capacity characterization be further investigated and potentially resolved for specific classes of source and channel distributions

The discontinuity behavior observed in the capacity characterization of communication systems with infinite alphabets can be further investigated and potentially resolved through targeted research and analysis. One approach to exploring this phenomenon is to conduct in-depth studies on specific classes of source and channel distributions to identify the underlying causes of discontinuities. By analyzing the characteristics of these distributions and their impact on capacity results, researchers can gain insights into the factors contributing to discontinuities and develop strategies to mitigate or eliminate them. Additionally, conducting simulation studies and numerical experiments can help in understanding the behavior of capacity bounds under different scenarios and conditions. By systematically varying parameters such as alphabet size, channel characteristics, and mutual information constraints, researchers can observe how discontinuities manifest and explore potential patterns or trends in the capacity results. This empirical approach can provide valuable empirical evidence to support theoretical findings and guide further investigations into discontinuity behavior. Furthermore, collaboration between information theorists, mathematicians, and communication engineers can facilitate interdisciplinary research efforts to address the discontinuity issue. By combining expertise from different fields, researchers can leverage diverse perspectives and methodologies to tackle the problem from multiple angles and develop comprehensive solutions. This collaborative approach can lead to innovative insights and novel techniques for resolving discontinuity challenges in capacity characterization for communication systems with infinite alphabets.

Are there any general patterns or underlying reasons that can explain the occurrence of discontinuities in capacity results for communication systems with infinite alphabets

The occurrence of discontinuities in capacity results for communication systems with infinite alphabets may exhibit general patterns or underlying reasons that can be explored to better understand this phenomenon. One potential general pattern could be related to the convergence properties of information measures and the behavior of capacity functions in the context of infinite alphabets. By analyzing the convergence behavior of entropy, mutual information, and capacity functions for specific classes of distributions, researchers may identify common trends or characteristics that lead to discontinuities in capacity results. Moreover, the discontinuity behavior could be linked to the complexity of the source and channel models considered in the capacity characterization. Certain types of distributions or channel models with specific properties or structures may inherently lead to discontinuities in capacity bounds due to their unique characteristics. By studying the impact of different distributional assumptions, channel conditions, and system constraints on capacity results, researchers can uncover the underlying reasons for discontinuities and develop tailored approaches to address them. Additionally, the discontinuity behavior in capacity results may be influenced by the interaction between source statistics, channel properties, and communication protocols. Variations in these factors can introduce discontinuities in capacity bounds, highlighting the intricate relationship between information theory, communication systems, and mathematical analysis. By investigating these interdependencies and their effects on capacity characterization, researchers can gain a deeper understanding of the origins of discontinuities and work towards resolving them systematically for specific classes of source and channel distributions.
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