Almost surely invariance principle for non-stationary and random intermittent dynamical systems

Su, Yaofeng

doi:10.3934/dcds.2019286

Cited by 7 publications

(2 citation statements)

References 18 publications

(75 reference statements)

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“…Another consequence of the martingale-coboundary representation is the ASIP, which in our context concerns almost sure approximation of the Birkhoff sum by Gaussians. The ASIP for random (and sequential) dynamical systems has been studied by several authors in recent years (see, for instance, [16, 20–22, 32, 50, 51]), and in this paper we will focus on the ASIP for Birkhoff sums generated by the skew product.…”

Section: Introduction and A Preview Of The Main Resultsmentioning

confidence: 99%

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

Hafouta

2023

Ergod. Th. Dynam. Sys.

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In this paper we show how to apply classical probabilistic tools for partial sums $\sum _{j=0}^{n-1}\varphi \circ \tau ^j$ generated by a skew product $\tau $ , built over a sufficiently well-mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $\varphi $ , we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate-deviations principle, several exponential concentration inequalities and Rosenthal-type moment estimates for skew products with $\alpha $ -, $\phi $ - or $\psi $ -mixing base maps and expanding-on-average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (in contrast to [2]) is that the random maps are not independent, they do not preserve the same measure and the observable $\varphi $ depends also on the base space. For stretched exponentially ${\alpha }$ -mixing base maps our proofs are based on multiple correlation estimates, which make the classical method of cumulants applicable. For $\phi $ - or $\psi $ -mixing base maps, we obtain an ASIP and maximal and concentration inequalities by establishing an $L^\infty $ convergence of the iterates ${\mathcal K}^{\,n}$ of a certain transfer operator ${\mathcal K}$ with respect to a certain sub- ${\sigma }$ -algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.

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Section: Introduction and A Preview Of The Main Resultsmentioning

confidence: 99%

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

Hafouta

2023

Ergod. Th. Dynam. Sys.

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“…For an RDS with a uniform almost sure mixing rate, quenched limit laws have been studied by numerous authors; see [ALS09, DFGTV18a, DFGTV18b, HL20, HS20, Su19a, Su19b]. However, it is quite natural and more likely to have a RDS (e.g.…”

Section: Introductionmentioning

confidence: 99%

Random Young towers and quenched limit laws

2022

Ergod. Th. Dynam. Sys.

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We obtain quenched almost sure invariance principles (with convergence rates) for random Young towers if the average measure of the tail of return times to the base of random towers decays sufficiently fast. We apply our results to some independent and identically distributed perturbations of some non-uniformly expanding maps. These imply that the random systems under study tend to a Brownian motion under various scalings.

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Loss of Memory and Moment Bounds for Nonstationary Intermittent Dynamical Systems

Korepanov

Leppänen

2021

Commun. Math. Phys.

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We study nonstationary dynamical systems formed by sequential concatenation of nonuniformly expanding maps with a uniformly expanding first return map. Assuming a polynomially decaying upper bound on the tails of first return times that is nonuniform with respect to location in the sequence, we derive a corresponding sharp polynomial rate of memory loss. As applications, we obtain new estimates on the rate of memory loss for random ergodic compositions of Pomeau-Manneville type intermittent maps and intermittent maps with unbounded derivatives.

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Almost surely invariance principle for non-stationary and random intermittent dynamical systems

Cited by 7 publications

References 18 publications

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

Random Young towers and quenched limit laws

Loss of Memory and Moment Bounds for Nonstationary Intermittent Dynamical Systems

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