[Paper Review] ε-Capacities and Second-Order Coding Rates for Channels with General State

Marco Tomamichel, Vincent Y. F. Tan|arXiv (Cornell University)|May 29, 2013

Wireless Communication Security Techniques29 references10 citations

TL;DR

This paper establishes ε-capacity and optimistic ε-capacity for state-dependent channels with causal state information at both encoder and decoder, enabling a characterization of the strong converse property. It derives second-order coding rates using refined concentration inequalities and Berry–Esséen theorems, with applications to i.i.d. and Markov states.

ABSTRACT

We consider state-dependent channels with general state available at both the encoder and the decoder. We establish the -capacity and the optimistic -capacity of such channels. The determination of these capacities allows us to provide a necessary and sufficient condition for the strong converse property to hold. We also provide a simpler sufficient condition on the firstand second-order statistics of the state process that ensures that the strong converse property holds. We then seek a finer characterization of these capacities in terms of second-order coding rates. The general results are supplemented by several examples including i.i.d. and Markov states and mixed channels. The proofs of the specializations of the general second-order result to the specific examples require new techniques such as multiple applications of various forms of the Berry-Esseen theorems.

Motivation & Objective

• To determine the ε-capacity and optimistic ε-capacity for channels with general state information available at both encoder and decoder.
• To provide necessary and sufficient conditions for the strong converse property to hold in such channels.
• To derive a simpler sufficient condition based on first- and second-order statistics of the state process.
• To offer a refined characterization of channel capacity through second-order coding rates.
• To extend general results to specific cases such as i.i.d. states, Markov states, and mixed channels.

Proposed method

• Theoretical derivation of ε-capacity and optimistic ε-capacity using information-theoretic tools for channels with general state.
• Application of concentration inequalities and large deviation techniques to analyze the strong converse property.
• Use of multiple forms of the Berry–Esséen theorem to establish second-order coding rate approximations.
• Development of new analytical techniques tailored to handle the statistical dependencies in i.i.d. and Markov state processes.
• Specialization of general second-order results to concrete channel models, including mixed channels.
• Proofs rely on asymptotic expansions and convergence bounds derived from central limit theorem refinements.

Experimental results

Research questions

• RQ1 What is the exact characterization of ε-capacity and optimistic ε-capacity for state-dependent channels with causal state information?
• RQ2 Under what conditions does the strong converse property hold for such channels?
• RQ3 How can second-order coding rates be derived for channels with general state processes?
• RQ4 What role do first- and second-order statistics of the state process play in determining the strong converse?
• RQ5 How do the general results specialize to i.i.d., Markov, and mixed channel models?

Key findings

• The ε-capacity and optimistic ε-capacity are fully characterized for channels with general state available at both encoder and decoder.
• A necessary and sufficient condition for the strong converse property is established, linking it to the structure of the state process.
• A simpler sufficient condition for the strong converse is derived based on the first- and second-order statistics of the state process.
• Second-order coding rates are obtained via refined applications of the Berry–Esséen theorem to model-specific cases.
• New proof techniques are developed to handle the statistical dependencies in i.i.d. and Markov state processes.
• The general second-order results are successfully specialized to mixed channels, demonstrating the framework's versatility.