Non-regular g-measures and variable length memory chains

TL;DR

This paper investigates the existence and uniqueness of stationary measures compatible with g-functions, especially in variable length memory chains, under new conditions that do not require vanishing uniform variation.

Contribution

It introduces new assumptions for variable length memory chains ensuring existence, uniqueness, and mixing properties without needing vanishing uniform variation.

Findings

Existence of compatible measures under null measure discontinuities.

Uniqueness and weak-Bernoullicity for variable length memory chains.

Discussion of non-essential discontinuities and related processes.

Abstract

It is well-known that there always exists at least one stationary measure compatible with a continuous g-function g. Here we prove that if the set of discontinuities of the g-function g has null measure under a candidate measure obtained by some asymptotic procedure, then this candidate measure is compatible with g. We explore several implications of this result, and discuss comparisons with the literature concerning assumptions and examples. Important part of the paper is concerned with the case of variable length memory chains, for which we obtain existence, uniqueness and weak-Bernoullicity (or $\beta$-mixing) under new assumptions. These results are specially designed for variable length memory models, and do not require vanishing uniform variation. We also provide a further discussion on some related notions, such as random context processes, non-essential discontinuities, and finally an example of everywhere discontinuous stationary measure.