Variable-Length Coding of Two-Sided Asymptotically Mean Stationary Measures

TL;DR

This paper investigates how variable-length coding affects the properties of two-sided infinite sequences, especially focusing on asymptotic mean stationarity and related measure-preserving conditions in a probabilistic framework.

Contribution

It provides new sufficient conditions under which variable-length coding preserves asymptotic mean stationarity and related measure-theoretic properties.

Findings

Conditions for preservation of asymptotic mean stationarity under coding

Relations between block entropies of original and coded processes

Construction of a stationary nonergodic process with specific properties

Abstract

We collect several observations that concern variable-length coding of two-sided infinite sequences in a probabilistic setting. Attention is paid to images and preimages of asymptotically mean stationary measures defined on subsets of these sequences. We point out sufficient conditions under which the variable-length coding and its inverse preserve asymptotic mean stationarity. Moreover, conditions for preservation of shift-invariant $\sigma$-fields and the finite-energy property are discussed and the block entropies for stationary means of coded processes are related in some cases. Subsequently, we apply certain of these results to construct a stationary nonergodic process with a desired linguistic interpretation.