Codes for Tasks and R\'enyi Entropy Rate

A task is randomly drawn from a finite set of tasks and is described using a fixed number of bits. All the tasks that share its description must be performed. Upper and lower bounds on the minimum $\rho$-th moment of the number of performed tasks are derived. The key is an analog of the Kraft Inequality for partitions of finite sets. When a sequence of tasks is produced by a source of a given R\'enyi entropy rate of order $1/(1+\rho)$ and $n$ tasks are jointly described using $nR$ bits, it is shown that for $R$ larger than the R\'enyi entropy rate, the $\rho$-th moment of the ratio of performed tasks to $n$ can be driven to one as $n$ tends to infinity, and that for $R$ less than the R\'enyi entropy rate it tends to infinity. This generalizes a recent result for IID sources by the same authors. A mismatched version of the direct part is also considered, where the code is designed according to the wrong law. The penalty incurred by the mismatch can be expressed in terms of a divergence measure that was shown by Sundaresan to play a similar role in the Massey-Arikan guessing problem.