We study the ε-capacity regions of mixed multiple-access channels (MACs) with general mixture and cost constraints by the information spectrum method. The starting point is a general formula for the ε-capacity region of general MACs with cost constraint, together with the random-coding and converse lemmas from which it is derived. For the quasi-static fading Gaussian MAC, with random fading coefficients and noise variance, we show that the ε-capacity region is the ε-outage capacity region, which gives the latter a Shannon-theoretic operational meaning. The direct part follows from the general formula, and the converse part from the strong converse property of the Gaussian MAC, which is stronger than the ε-strong converse, applied to each component. The ε-capacity regions coincide in all four scenarios of availability of channel state information (CSI), and the results extend to K users. With CSI at both terminals, variable-rate coding enlarges the 0-capacity region to the polymatroid whose rank function is the average of those of the components. For mixed MACs with general components and a countable parameter space, CSI at the receiver alone never enlarges the ε-capacity region, and the region with CSI at the transmitters is given by an information spectrum formula; unlike the Gaussian case, the regions with and without CSI at the transmitters may differ. For uncountable parameter spaces, only the formula for the region with CSI at both terminals survives in general. Parts II and III treat finite-alphabet mixed MACs and the strong converse for the K-user Gaussian MAC, respectively.
This paper, Part III of a series, treats $K$-user multiple-access channels (MACs). Its starting point is a pair of fundamental lemmas, a random-coding lemma of Feinstein type and a converse lemma of Verdú--Han type, which are the $K$-user extensions of the two lemmas underlying Han's information spectrum approach to the general two-user MAC. From them we derive the capacity region of the general $K$-user MAC with cost constraint and show that it contains the polymatroid capacity region of the $K$-user Gaussian MAC. We then prove, by an elementary information spectrum calculation that adapts a diffuse--exceptional subspace decomposition used in recent second-order converses for the Gaussian MAC, that for every subset $\cS$ of users and every independent input satisfying the power constraints, the conditional information spectral-sup rate of the $K$-user Gaussian MAC does not exceed $\frac12\log(1+P_\cS/N)$, where $P_\cS$ is the total power of the users in $\cS$, with a probability bound that does not depend on the input. Consequently, the strong converse for the $K$-user Gaussian MAC, in the sense that the error probability tends to one for every rate tuple outside the capacity region, follows directly from the converse lemma, without the wringing technique and without hypothesis-testing meta-converses. This strong converse was first established by Fong and Tan, whose per-blocklength bounds yield it, through wringing and quantization. It is the key ingredient in the converse parts of the coding theorems for mixed Gaussian MACs in Part I, where the $\varepsilon$-strong converse would not suffice.
In a nonergodic channel such as a slowly fading link, the channel law is drawn once and held fixed for the whole codeword, and the relevant performance measure is the $\varepsilon$-capacity. We study the $\varepsilon$-capacity regions of two-user mixed multiple-access channels (MACs) with finite alphabets, additive cost constraints and an arbitrary mixing measure. The direct parts are derived from a single information spectrum formula; the converse parts use, in addition, Fano's inequality and Ahlswede's strong converse applied to each component. Without knowledge of the realized component, we prove a single-letter inner bound for every $0\le\varepsilon<1$, in which one input serves all components; for $\varepsilon=0$ it is tight, and the capacity region is given by the essential infima of the component mutual informations. We prove a single-letter outer bound, the outage region, in which each component may use its own input; it holds with or without channel state information (CSI) at either terminal, and it is the $\varepsilon$-capacity region when the encoders know the component and the parameter space is countable. Both bounds coincide, in all four CSI scenarios, for well-ordered mixed MACs, which include degradedly ordered families and mixtures with a common optimal input, such as mixed additive MACs. Without CSI the outer bound can be strictly loose, and the no-CSI $\varepsilon$-capacity region for $0<\varepsilon<1$ remains open in general. Finally, we extend Ahlswede's strong converse to $K$ users with cost constraints and show that most of these results extend to $K$-user mixed MACs.