Abstract
Regular $g$-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist $g$-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of processes with rather nice attributes: It is a chain with variable-length memory, characterized by the absence of phase coexistence and the existence of a visible renewal scheme
Highlights
Measures on EZ are the object of two very developed theories
Regular g-measures are discrete-time processes determined by conditional expectations with respect to the past
We present a simple example settling this issue in a negative way: there exist g-measures that are continuous and non-null but are not Gibbsian
Summary
Measures on EZ are the object of two very developed theories. On the one hand, the theory of chains of complete connections, started in [16] and developed under a variety of names in slightly non-equivalent frameworks: chains of infinite order [10], g-measures [12], uniform martingales [11], etc. The previous questions should be stated, in the common non-null framework In this set-up, the equivalence of both theories has long been known to be true for Markov processes and fields (see, for instance, [9, Chapter 11]) and when continuity rates are exponentially decreasing [6]. The measure μ is non-null and consistent with a continuous g-function This means that, upon conditioning, Electronic Communications in Probability it becomes asymptotically insensitive to the far past. The Gibbsianness question refers to whether conditional probabilities converge (to whatever) as both the first 1 to the left and the first 1 to the right move away This is an essential property in the sense that its absence can not be fixed by measure-zero redefinitions. The measure μ is, almost Gibbsian and weakly Gibbsian (see, for instance, [5, Section 4.4] for the corresponding definitions and historical references to these notions)
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