On probability measures specified by regular g-functions (2008)


We consider measures on $\{\pm 1\}^\Z$ for which the conditional probability that the spin at $n+1$ takes the value $+1$, given the values of all spins at sites $n, n-1,\dots$, is specified by some a priori given function $g$. We first show how the regularity of $g$ (continuity and uniform non-nullness) allows to construct explicitely measures with such dependencies. We then consider the problem of extreme decomposition for the set of measures specified by $g$. Finally, we expose the uniqueness result of Johansson and \"Oberg \cite{JoOb}: when the variation of $g$ is $\ell^2$-summable, then there exists a unique invariant measure.

Keywords: regular g-function, variation, stationary process, extreme decomposition, uniqueness criterium.

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