Optimistic Shannon coding theorems for arbitrary single-user systems

The conventional definitions of the source coding rate and of channel capacity require the existence of reliable codes for all sufficiently large block lengths. Alternatively, if it is required that good codes exist for infinitely many block lengths, then optimistic definitions of source coding rate and channel capacity are obtained. In this work, formulas for the optimistic minimum achievable fixed-length source coding rate and the minimum ε-achievable source coding rate for arbitrary finite-alphabet sources are established. The expressions for the optimistic capacity and the optimistic ε-capacity of arbitrary single-user channels are also provided. The expressions of the optimistic source coding rate and capacity are examined for the class of information stable sources and channels, respectively. Finally, examples for the computation of optimistic capacity are presented.