Explicit estimates in the Bramson–Kalikow model

Gallesco, Christophe; Gallo, Sandro; Takahashi, Daniel Yasumasa

doi:10.1088/0951-7715/27/9/2281

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…The sequence of measures µ [m k ] , k ≥ 1 was used by [23] for their proof of phase transition of the Bramson-Kalikow model. Suppose for now that there exists a shift-invariant measure µ compatible with g. We can easily derive a lower bound for d(µ, µ [m k ] ).…”

Section: Explicit Upper Bound For the D-distance And Speed Of Markovi...mentioning

confidence: 99%

“…One of the main interests in SCUMs comes from the fact that they exhibit different mixing properties depending on the characteristics of the probability kernels. For instance, kernels with strong dependence on the past can have two or more shift invariant measures compatible with the kernel [4,33,23,22,14,1]. Weak dependence on the past leads to uniqueness of the compatible measure.…”

Section: Introductionmentioning

confidence: 99%

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Gaussian concentration bounds for stochastic chains of unbounded memory

Chazottes,

Gallo,

Takahashi

2023

Ann. Appl. Probab.

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We obtain optimal Gaussian concentration bounds (GCBs) for stochastic chains of unbounded memory (SCUMs) on countable alphabets. These stochastic processes are also known as "chains with complete connections" or "g-measures". We consider two different conditions on the kernel: (1) when the sum of its oscillations is less than one, or (2) when the sum of its variations is finite, i.e., belongs to 1 (N). We also obtain explicit constants as functions of the parameters of the model. The proof is based on maximal coupling. Our conditions are optimal in the sense that we exhibit examples of SCUMs that do not have GCB and for which the sum of oscillations is strictly larger than one, or the variation belongs to 1+ (N) for any > 0. These examples are based on the existence of phase transitions. We also extend the validity of GCB to a class of functions which can depend on infinitely many coordinates.We illustrate our results by three applications. First, we derive a Dvoretzky-Kiefer-Wolfowitz type inequality which gives a uniform control on the fluctuations of the empirical measure. Second, in the finite-alphabet case, we obtain an upper bound on the d-distance between two stationary SCUMs and, as a by-product, we obtain new (explicit) bounds on the speed of Markovian approximation in d. Third, we obtain exponential rate of convergence for Birkhoff sums of a certain class of observables.

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Section: Explicit Upper Bound For the D-distance And Speed Of Markovi...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gaussian concentration bounds for stochastic chains of unbounded memory

Chazottes,

Gallo,

Takahashi

2023

Ann. Appl. Probab.

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Uniqueness vs. non-uniqueness for complete connections with modified majority rules

Dias

Friedli

2015

Probab. Theory Relat. Fields

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Abstract. We take a closer look at a class of chains with complete connections inspired by the one of Berger, Hoffman and Sidoravicius [1]. Besides giving a sharper description of the uniqueness and non-uniqueness regimes, we show that if the pure majority rule used to fix the dependence on the past is replaced with a function that is Lipschitz at the origin, then uniqueness always holds, even with arbitrarily slow decaying variation.

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