Two Measures of Dependence

Two families of dependence measures between random variables are introduced. They are based on the R\'enyi divergence of order $\alpha$ and the relative $\alpha$-entropy, respectively, and both dependence measures reduce to Shannon's mutual information when their order $\alpha$ is one. The first measure shares many properties with the mutual information, including the data-processing inequality, and can be related to the optimal error exponents in composite hypothesis testing. The second measure does not satisfy the data-processing inequality, but appears naturally in the context of distributed task encoding.Comment: 40 pages; 1 figure; published in Entrop

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.Comment: 5 pages, to be presented at ISIT 2014; minor changes in the presentation, added a referenc

On Multipath Fading Channels at High SNR

This paper studies the capacity of discrete-time multipath fading channels. It is assumed that the number of paths is finite, i.e., that the channel output is influenced by the present and by the L previous channel inputs. A noncoherent channel model is considered where neither transmitter nor receiver are cognizant of the fading's realization, but both are aware of its statistic. The focus is on capacity at high signal-to-noise ratios (SNR). In particular, the capacity pre-loglog - defined as the limiting ratio of the capacity to loglog SNR as SNR tends to infinity - is studied. It is shown that, irrespective of the number paths L, the capacity pre-loglog is 1.Comment: To be presented at the 2008 IEEE Symposium on Information Theory (ISIT), Toronto, Canada; replaced with version that appears in the proceeding