Average redundancy of the Shannon code for Markov sources

TL;DR

This paper extends the understanding of Shannon code redundancy from memoryless sources to Markov sources, revealing conditions for convergence or oscillation in redundancy behavior using advanced analytic methods.

Contribution

It provides the first detailed characterization of redundancy behavior for Markov sources, including conditions for convergence and oscillation, using spectral and Fourier analysis.

Findings

Redundancy exhibits either convergence or oscillation depending on source parameters.

The behavior is characterized for irreducible, periodic, and aperiodic Markov sources.

Analytic methods like Fourier series and spectral analysis are employed.

Abstract

It is known that for memoryless sources, the average and maximal redundancy of fixed-to-variable length codes, such as the Shannon and Huffman codes, exhibit two modes of behavior for long blocks. It either converges to a limit or it has an oscillatory pattern, depending on the irrationality or rationality, respectively, of certain parameters that depend on the source. In this paper, we extend these findings, concerning the Shannon code, to the case of a Markov source, which is considerably more involved. While this dichotomy, of convergent vs. oscillatory behavior, is well known in other contexts (including renewal theory, ergodic theory, local limit theorems and large deviations of discrete distributions), in information theory (e.g., in redundancy analysis) it was recognized relatively recently. To the best of our knowledge, no results of this type were reported thus far for Markov sources. We provide a precise characterization of the convergent vs. oscillatory behavior of the Shannon code redundancy for a class of irreducible, periodic and aperiodic, Markov sources. These findings are obtained by analytic methods, such as Fourier/Fejer series analysis and spectral analysis of matrices.