Non-regular g-measures and variable length memory chains

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 a 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 dedicated to the case of variable length memory chains, for which we obtain existence, uniqueness and weak-Bernoullicity (or β-mixing) under new assumptions. These results are specially designed for variable length memory models, and do not require uniform continuity. We also provide a further discussion on some related notions, such as random context processes, non-essential discontinuities and everywhere discontinuous stationary measures.

Number of visits in arbitrary sets for $ϕ$-mixing dynamics

“…Then by the law of large numbers for the product measure µ, we have that ν(G) = 1. It is proved in Ferreira et al (2020)…”

“…Technically, the discontinuity of ḡ at 0 ∞ is an essential discontinuity, borrowing the terminology used in the context of statistical physics (Fernández, 2005). This is a discontinuity which cannot be removed by changing function ḡ on a null μ-measure subset of Σ (Ferreira et al, 2020).…”

“…, the product of two consecutive coordinates. The measure ν := µ • Π −1 has a g-function which is essentially discontinuous everywhere (Verbitskiy, 2015;Ferreira et al, 2020). Let us now write down its g-function g .…”

Markov and almost Markov properties in one, two or more directions

“…But wilder things may happen. There exists a positive g-function (everywhere discontinuous) with no g-measure; an example, due to Noam Berger, is given in [38]. In the same paper is also exibited a g-measure which is proved to be everywhere essentially discontinuous; this example has first appeared in Harry Furstenberg's thesis and has been published in [48].…”

“…The smallness of this set can be measured in several ways: by its shape (viewed as a set of infinite paths) as in [50] or by assuming its topological pressure to be strictly negative (it mixes the shape of the set with the values g takes on this set) as in [51]. More generally, the existence is proved to hold in [38] if the set of discontinuity points is being given zero measure for a dynamically defined candidate measure. But wilder things may happen.…”