On non-regular<i>g</i>-measures

Gallo, Sandro; Paccaut, Frédéric

doi:10.1088/0951-7715/26/3/763

Cited by 17 publications

(40 citation statements)

References 16 publications

(33 reference statements)

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“…For instance, Comets et al [20] have shown that, if a inf w p(a|w) ≥ δ and ct(k) ≤ C k −(1+α) for constants C, α, δ > 0, then the process X is φ-mixing with polynomial rate ≈ k −α . On the other hand, there seem to be no known examples of processes that are not a.s. continuous [17], so "ct(X −1 −k ) → 0 almost surely" (and non-uniformly) is compatible with many different kinds of mixing hypotheses.…”

Section: Remark 2 (Relationship To Continuity and Mixing Properties):mentioning

confidence: 81%

Stochastic Processes With Random Contexts: A Characterization and Adaptive Estimators for the Transition Probabilities

Oliveira

2015

IEEE Trans. Inform. Theory

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This paper introduces the concept of random context representations for the transition probabilities of a finite-alphabet stochastic process. Processes with these representations generalize context tree processes (a.k.a. variable length Markov chains), and are proven to coincide with processes whose transition probabilities are almost surely continuous functions of the (infinite) past. This is similar to a classical result by Kalikow about continuous transition probabilities. Existence and uniqueness of a minimal random context representation are shown, in the sense that there exists a unique representation that "looks into the past as little as possible" in order to determine the next symbol. Both this representation and the transition probabilities can be consistently estimated from data, and some finite sample adaptivity properties are also obtained (including an oracle inequality). In particular, the estimator achieves minimax performance, up to logarithmic factors, for the class of binary renewal processes whose arrival distributions have bounded moments of order 2 + γ.

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Section: Remark 2 (Relationship To Continuity and Mixing Properties):mentioning

confidence: 81%

Stochastic Processes With Random Contexts: A Characterization and Adaptive Estimators for the Transition Probabilities

Oliveira

2015

IEEE Trans. Inform. Theory

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“…s A(α) * E A(α)A s ) = Tr(A(α) * E A(α)I) = ν(α),so ν is consistent with extensions to the left. Similarly,(3) shows that s ν(αs) = ν(α).…”

mentioning

confidence: 84%

Ergodic theory of Kusuoka measures

Johansson

Öberg²,

Pollicott³

2017

J. Fractal Geom.

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In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role (cf.[13], [7], [2]). Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the Hölder continuous functions, as well as the highly discontinuous g-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpiński gaskets.

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“…They constitute an important class of stochastic models, which includes, for example, Markov chains, stochastic models that exhibit nonuniqueness and models that are not Gibbsian (Fernández et al, 2011). The question of uniqueness of g-measures was extensively studied and important progresses have been obtained in several areas related to probability and ergodic theory, from the seminal works of Onicescu & Mihoc (1935); Doeblin & Fortet (1937) to recent advances in Johansson et al (2012); Gallo & Paccaut (2013), and the contributions of Harris (1955); Keane (1972); Walters (1975); Lalley (1986); Stenflo (2003); Fernández & Maillard (2005) among many others. Notwithstanding, the problem of non-uniqueness is much less understood and the literature is still based on few examples (Bramson & Kalikow, 1993;Hulse, 2006;Berger et al, 2005).…”

Section: Introductionmentioning

confidence: 99%

Explicit estimates in the Bramson–Kalikow model

Gallesco¹,

Gallo²,

Takahashi³

2014

Nonlinearity

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Abstract. The aim of the present article is to explicitly compute parameters for which the Bramson-Kalikow model exhibits phase-transition. The main ingredient of the proof is a simple new criterion for non-uniqueness of g-measures. We show that the existence of multiple g-measures compatible with a function g can be proved by estimating thed-distances between some suitably chosen Markov chains. The method is optimal for the important class of binary regular attractive functions, which includes the Bramson-Kalikow model.

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On non-regularg-measures

Cited by 17 publications

References 16 publications

Stochastic Processes With Random Contexts: A Characterization and Adaptive Estimators for the Transition Probabilities

Stochastic Processes With Random Contexts: A Characterization and Adaptive Estimators for the Transition Probabilities

Ergodic theory of Kusuoka measures

Explicit estimates in the Bramson–Kalikow model

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