Non-regular g-measures and variable length memory chains
arXiv:1907.02442 · doi:10.1088/1361-6544/aba0c5
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 -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.
arXiv comment33 pages
References in corpus (11)
- Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions
- Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
- Context tree selection and linguistic rhythm retrieval from written texts
- Entropic repulsion and lack of the -measure property for Dyson models
- Variable length Markov chains and dynamical sources
- On non-regular g-measures
- Conditional ergodicity in infinite dimension
- Quadratic Semimartingale BSDEs under an Exponential Moments Condition
- Characterization of stationary probability measures for Variable Length Markov Chains
- Stochastic processes with random contexts: a characterization, and adaptive estimators for the transition probabilities
- Persistent random walks