Large deviations and central limit theorems for sequential and random systems of intermittent maps

Nicol, Matthew; Pereira, Felipe Perez; Torok, Andrew

doi:10.48550/arxiv.1909.07435

Cited by 3 publications

(7 citation statements)

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“…As in [19], we prove concentration bounds not only for Birkhoff sums, but for a more general class of separately Lipschitz (or separately Hölder) functions on [0, 1] N , see Theorem 3.11 and Remark 3.3. Theorem 1.2 improves the moment bounds in Nicol, Pereira and Török [33] and Su [41], and implies the following bounds on large and moderate deviations: Corollary 2.1. In the notation of Theorem 1.2, for every p > 2,…”

Section: Discussionmentioning

confidence: 56%

“…Also, we found that for γ * ∈ (0, 1/2), Theorem 1.2 can be proved using memory loss with asymptotics O(n −1/γ * +1 ) as in Theorem 1.1(a), and close to optimal results can be obtained with the slightly weaker bound O n −1/γ * +1 (log n) 1/γ * from [1], as it is done in [33]. For γ * ∈ (1/2, 1) the situation is significantly more complicated.…”

Section: 3mentioning

confidence: 58%

“…for each p > max{1, 1/γ * − 1} from [33,Theorem 4.1]. We remove the logarithmic term, get a better power of ε when γ * ∈ (1/2, 1) and allow the observables v n to depend on n.…”

Section: Discussionmentioning

confidence: 99%

“…Alternatively, one can use the moment bounds from [33] or [41], but these were not available when we started this project.…”

Section: 3mentioning

confidence: 99%

“…We study limit laws for nonstationary dynamical systems, a topic of very intense recent interest, see [2,3,4,5,9,13,14,15,16,20,22,23,29,33,34,38,40,41] and more.…”

Section: Introductionmentioning

confidence: 99%

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Loss of memory and moment bounds for nonstationary intermittent dynamical systems

Korepanov,

Leppänen

2020

Preprint

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Section: Discussionmentioning

confidence: 56%

Section: 3mentioning

confidence: 58%

“…for each p > max{1, 1/γ * − 1} from [33,Theorem 4.1]. We remove the logarithmic term, get a better power of ε when γ * ∈ (1/2, 1) and allow the observables v n to depend on n.…”

Section: Discussionmentioning

confidence: 99%

“…Alternatively, one can use the moment bounds from [33] or [41], but these were not available when we started this project.…”

Section: 3mentioning

confidence: 99%

“…We study limit laws for nonstationary dynamical systems, a topic of very intense recent interest, see [2,3,4,5,9,13,14,15,16,20,22,23,29,33,34,38,40,41] and more.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Loss of memory and moment bounds for nonstationary intermittent dynamical systems

Korepanov,

Leppänen

2020

Preprint

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Random composition of L-S-V maps sampled over large parameter ranges

2021

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Liverani–Saussol–Vaienti (L–S–V) maps form a family of piecewise differentiable dynamical systems on [0, 1] depending on one parameter ω ∈ R + . These maps are everywhere expanding apart from a neutral fixed point. It is well known that depending on the amount of expansion close to the neutral point, they have either an absolutely continuous invariant probability measure and polynomial decay of correlations (ω < 1), or a unique physical measure that is singular and concentrated at the neutral point (ω > 1). In this paper, we study the composition of L–S–V maps whose parameters are randomly sampled from a range in R + , and where these two contrasting behaviours are mixed. We show that if the parameters ω < 1 are sampled with positive probability, then the stationary measure of the random system is absolutely continuous; the annealed decay rate of correlations is close (or in some cases equal) to the fastest rate of decay among those of the sampled systems; and suitably rescaled Birkhoff averages converge to limit laws. In contrast to previous studies where ω ∈ [0, 1], we allow ω > 1 in our sampling distribution. We also show that one can obtain similar decay of correlation rates for ω ∈ R + , when sampling is done with respect to a family of smooth, heavy-tailed distributions.

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Random Composition of L-S-V Maps Sampled Over Large Parameter Ranges

Bose,

Quas,

Tanzi

2020

Preprint

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Liverani-Saussol-Vaienti (L-S-V) maps form a family of piecewise differentiable dynamical systems on [0, 1] depending on one parameter ω ∈ R + . These maps are everywhere expanding apart from a neutral fixed point. It is well known that depending on the amount of expansion close to the neutral point, they have either an absolutely continuous invariant probability measure and polynomial decay of correlations (ω < 1), or a unique physical measure that is singular and concentrated at the neutral point (ω > 1). In this paper, we study the composition of L-S-V maps whose parameters are randomly sampled from a range in R + , and where these two contrasting behaviours are mixed. We show that if the parameters ω < 1 are sampled with positive probability, then the stationary measure of the random system is absolutely continuous; the annealed decay rate of correlations is close (or in some cases equal) to the fastest rate of decay among those of the sampled systems; and suitably rescaled Birkhoff averages converge to limit laws. In contrast to previous studies where ω ∈ [0, 1], we allow ω > 1 in our sampling distribution. We also show that one can obtain similar decay of correlation rates for ω ∈ [0, ∞), when sampling is done with respect to a family of smooth, heavy-tailed distributions.

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Large deviations and central limit theorems for sequential and random systems of intermittent maps

Cited by 3 publications

References 0 publications

Loss of memory and moment bounds for nonstationary intermittent dynamical systems

Loss of memory and moment bounds for nonstationary intermittent dynamical systems

Random composition of L-S-V maps sampled over large parameter ranges

Random Composition of L-S-V Maps Sampled Over Large Parameter Ranges

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