Semi-continuous arbitrarily varying channels with general state constraints (Q1373104)

This paper is a continuation of the author's earlier paper [\textit{Bui Van Thanh}, The capacity of arbitrarily varying channels under general state constraints, Probl. Control Inf. Theory 19, 151-165 (1990; Zbl 0706.94008)], where the concept of capacity of arbitrary varying channels under general state constraints was introduced. The state constraints were expressed in terms of types of state sequences. The problem rose from earlier papers of \textit{I. Csiszár} and \textit{P. Narayan} [IEEE Trans. Inform. Theory IT-34, 27-34 (1988; Zbl 0649.94009); ibid. 181-193 (1988; Zbl 0652.94005)], (cited incorrectly in the references), who considered the state constraints given in terms of a function. The AVC's having continuous alphabets and set of states are the most important, but they are relatively less understood. In this paper the author extends the results of his paper cited above, where he gave an exact formula for the capacity of discrete AVC's with finite state sets, to memoryless semi-continuous AVC's, that is, the AVC's with finite alphabet \({\mathcal X}\) and general output alphabet \({\mathcal Y}\) and state set \({\mathcal S}\). Dropping the assumption on finiteness of \({\mathcal S}\) and \({\mathcal Y}\) presents no difficulties. Dealing with infinite \({\mathcal X}\) is more difficult and to get satisfactory results for that case one needs strong regularity assumptions. This is not surprising because for infinite input alphabets, no general solution is known, even to the simpler compound channel capacity problem.