Sunklodas’ Approach to Normal Approximation for Time-Dependent Dynamical Systems

Leppänen, Juho; Stenlund, Mikko

doi:10.1007/s10955-020-02636-7

Cited by 5 publications

(8 citation statements)

References 66 publications

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“…A similar condition was introduced by Leppänen in [Lep17] and further studied by Leppänen and Stenlund in [LS17,LS20]. In particular, [Lep17] showed that functional correlation decay implies a multi-dimensional CLT with bounds on the rate of decay.…”

Section: Resultsmentioning

confidence: 68%

Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-Mixing Nonuniformly Hyperbolic Maps

Fleming-Vázquez

2022

Commun. Math. Phys.

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Consider a nonuniformly hyperbolic map $$ T:M\rightarrow M $$ T : M → M modelled by a Young tower with tails of the form $$ O(n^{-\beta }) $$ O ( n - β ) , $$ \beta >2 $$ β > 2 . We prove optimal moment bounds for Birkhoff sums $$ \sum _{i=0}^{n-1}v\circ T^i $$ ∑ i = 0 n - 1 v ∘ T i and iterated sums $$ \sum _{0\le i<j<n}v\circ T^i\, w\circ T^j $$ ∑ 0 ≤ i < j < n v ∘ T i w ∘ T j , where $$ v,w:M\rightarrow {{\mathbb {R}}} $$ v , w : M → R are (dynamically) Hölder observables. Previously iterated moment bounds were only known for $$ \beta >5$$ β > 5 . Our method of proof is as follows; (i) prove that $$ T$$ T satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimates. Such iterated moment bounds arise when using rough path theory to prove deterministic homogenisation results. Indeed, by a recent result of Chevyrev, Friz, Korepanov, Melbourne & Zhang we have convergence to an Itô diffusion for fast-slow systems of the form $$\begin{aligned} x^{(n)}_{k+1}=x_k^{(n)}+n^{-1}a(x_k^{(n)},y_k)+n^{-1/2}b(x_k^{(n)},y_k) , \quad y_{k+1}=Ty_k \end{aligned}$$ x k + 1 ( n ) = x k ( n ) + n - 1 a ( x k ( n ) , y k ) + n - 1 / 2 b ( x k ( n ) , y k ) , y k + 1 = T y k in the optimal range $$ \beta >2$$ β > 2 .

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Section: Resultsmentioning

confidence: 68%

Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-Mixing Nonuniformly Hyperbolic Maps

Fleming-Vázquez

2022

Commun. Math. Phys.

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“…ω ∈ Ω and φ ∈ BV . The conclusion of the lemma now follows readily from (37) and the triangle inequality.…”

Section: Twisted Transfer Operatorsmentioning

confidence: 78%

“…Among many remarkable contributions, we particularly emphasize those dealing with the decay of correlations [2,6,7,9,12,15], various (quenched or annealed) limit laws [1,4,16,17,18,19,20,30,31,32,39,43,44,48], as well as recent results devoted to the linear response of random dynamical systems [8,23,47]. For similar results in the closely related context of sequential dynamical systems, we refer to [10,11,33,35,36,37,29,42] and references therein.…”

Section: Introductionmentioning

confidence: 98%

Quenched limit theorems for expanding on average cocycles

Dragičević¹,

Sedro²

2021

Preprint

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We prove quenched versions of a central limit theorem, a large deviations principle as well as a local central limit theorem for expanding on average cocycles. This is achieved by building an appropriate modification of the spectral method for nonautonomous dynamics developed by Dragičević et al. (Comm Math Phys 360: 1121-1187, to deal with the case of random dynamics that exhibits nonuniform decay of correlations, which are ubiquitous in the context of the multiplicative ergodic theory. Our results provide an affirmative answer to a question posed by Buzzi (Comm Math Phys 208: 25-54, 1999).

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“…for P-a.e. ω ∈ Ω, where t T denotes the transpose of t. Substituting θ = t/ √ n in (33) and taking into account…”

Section: Central Limit Theoremmentioning

confidence: 99%

“…Furthermore, we mention the recent interesting papers by Bahsoun and collaborators [2,4,5] as well as Su [43] concerned with the decay of correlation and limit laws for systems which can be modelled by random Young towers. Further relevant contributions to the study of statistical properties of random or time-dependent dynamics have been established by Nándori,Szász,and Varjú [38], Nicol, Török and Vaienti [39], Hella and Stenlund [28], Leppänen and Stenlund [32,33] as well as the second author [25,26].…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for random expanding or hyperbolic dynamical systems and vector-valued observables

Dragičević,

Hafouta

2020

Preprint

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The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author et al. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for vector-valued observables. We stress that the previous works considered exclusively the case of scalar-valued observables.In another direction, we show that this method can be used to establish a variety of new limit laws (either for scalar or vector-valued observables) that have not been discussed previously in the literature for the classes of dynamics we consider. More precisely, we establish the moderate deviation principle, concentration inequalities, Berry-Esseen estimates as well as Edgeworth and large deviation expansions.Although our techniques rely on the approach developed in the previous works of the first author et al., we emphasize that our arguments require several nontrivial adjustments as well as new ideas.

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Sunklodas’ Approach to Normal Approximation for Time-Dependent Dynamical Systems

Cited by 5 publications

References 66 publications

Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-Mixing Nonuniformly Hyperbolic Maps

Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-Mixing Nonuniformly Hyperbolic Maps

Quenched limit theorems for expanding on average cocycles

Limit theorems for random expanding or hyperbolic dynamical systems and vector-valued observables

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