Dynamic uniqueness for stochastic chains with unbounded memory |
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Authors: | Christophe Gallesco Sandro Gallo Daniel Y. Takahashi |
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Affiliation: | 1. Departmento de Estatística, Instituto de Matemática, Estatística e Ciência de Computação, Universidade de Campinas, Brazil;2. Departamento de Estatística, Universidade Federal de São Carlos, Brazil;3. Princeton Neuroscience Institute, Princeton University, USA |
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Abstract: | We say that a probability kernel exhibits dynamic uniqueness (DU) if all the stochastic chains starting from a fixed past coincide on the future tail -algebra. Our first theorem is a set of properties that are pairwise equivalent to DU which allow us to understand how it compares to other more classical concepts. In particular, we prove that DU is equivalent to a weak- summability condition on the kernel. As a corollary to this theorem, we prove that the Bramson–Kalikow and the long-range Ising models both exhibit DU if and only if their kernels are summable. Finally, if we weaken the condition for DU, asking for coincidence on the future -algebra for almost every pair of pasts, we obtain a condition that is equivalent to -mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak- summability condition on the kernel is equivalent to the -mixing of the compatible stationary chain. |
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Keywords: | primary 60G10 secondary 60G99 Stochastic chains with unbounded memory Phase transition Coupling Bramson–Kalikow Total variation distance |
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