Information Sources on a Bratteli diagram

TL;DR

This paper studies information sources on Bratteli diagrams, establishing fundamental theorems like ergodic decomposition, entropy rate, and source coding, using Vershik transformation, generalizing finite alphabet sources.

Contribution

It introduces a framework for analyzing information sources on Bratteli diagrams, extending classical results to this more general setting with new theorems and proof techniques.

Findings

Ergodic decomposition for sources on Bratteli diagrams

Entropy rate theorems for these sources

Source coding theorems including lossless and lossy cases

Abstract

A Bratteli diagram is a type of graph in which the vertices are split into finite subsets occupying an infinite sequence of levels, starting with a bottom level and moving to successively higher levels along edges connecting consecutive levels. An information source on a Bratteli diagram consists of a sequence of PMFs on the vertex sets at each level that are compatible under edge transport. By imposing a regularity condition on the Bratteli diagram, we obtain various results for its information sources including ergodic and entropy rate decomposition theorems, a Shannon-Mcmillan-Breiman theorem, and lossless and lossy source coding theorems. Proof methodology exploits the Vershik transformation on the path space of a Bratteli diagram. Some results for finite alphabet stationary sequential information sources are seen to be a special case of the results of this paper.