 Dynamic uniqueness for stochastic chains with unbounded memoryChristophe Gallesco, Sandro Gallo and Daniel Y. Takahashi Stochastic Processes and their Applications, 2018, vol. 128, issue 2, 689-706 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 σ-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-ℓ2 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 ℓ2 summable. Finally, if we weaken the condition for DU, asking for coincidence on the future σ-algebra for almost every pair of pasts, we obtain a condition that is equivalent to β-mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak-ℓ2 summability condition on the kernel is equivalent to the β-mixing of the compatible stationary chain. Keywords: Stochastic chains with unbounded memory; Phase transition; Coupling; β-mixing; Bramson–Kalikow; Total variation distance (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:2:p:689-706 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spa.2017.06.004 Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
Is your work missing from RePEc? Here is how to contribute.
Questions or problems? Check the EconPapers FAQ or send mail to .
EconPapers is hosted by the
School of Business at Örebro University.