The Renyi Capacity and Center

Abstract

Renyi's information measures ---the Renyi information, mean, capacity, radius, and center--- are analyzed relying on the elementary properties of the Renyi divergence and the power means. The van Erven-Harremoes conjecture is proved for any positive order and for any set of probability measures on a given measurable space and a generalization of it is established for the constrained variant of the problem. The finiteness of the order $\alpha$ Renyi capacity is shown to imply the continuity of the Renyi capacity on $(0,\alpha ]$ and the uniform equicontinuity of the Renyi information, both as a family of functions of the order indexed by the priors and as a family of functions of the prior indexed by the orders. The Renyi capacities and centers of various families of Poisson processes are derived as examples.

32+8 pages. An error in the statement of Lemma 8 is corrected. The change is inconsequential for the rest of the paper

References in corpus (3)

• Fundamental bound on the reliability of quantum information transmission
• On the strong converses for the quantum channel capacity theorems
• The Augustin Center and The Sphere Packing Bound For Memoryless Channels

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• Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions
• The Augustin Capacity and Center
• Properties of Noncommutative Renyi and Augustin Information
• A Scale-invariant Generalization of the Rényi Entropy, Associated Divergences and their Optimizations under Tsallis' Nonextensive Framework
• The Sphere Packing Bound For Memoryless Channels
• Characterisation of quantum betting tasks in terms of Arimoto mutual information
• Conditional Rényi entropy and the relationships between Rényi capacities
• On the Existence of the Augustin Mean
• The Sphere Packing Bound for DSPCs with Feedback a la Augustin
• The Mutual Information In The Vicinity of Capacity-Achieving Input Distributions