Speed of convergence for laws of rare events and escape rates

Freitas, Ana Cristina Moreira; Freitas, Jorge Milhazes; Todd, Michael J.

doi:10.1016/j.spa.2014.11.011

Cited by 35 publications

(41 citation statements)

References 43 publications

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“…On the other hand, EVLs for the partial maximum of dynamically defined stochas- tic processes is a much recent topic and have been proved directly in the recent papers [72,134,74,137,138,139,49,141,140,142,207,144,143]. We highlight the pioneering work of Collet [72] for the innovative ideas introduced.…”

Section: Non-exponential Lawsmentioning

confidence: 90%

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Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

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194

253

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ii 4.1.2 The New Conditions 44 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 46 4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 48 4.2 Extreme Values for Dynamically Defined Stochastic Processes 53 4.2.1 Observables and Corresponding Extreme Value Laws 55 4.2.2 Extreme Value Laws for Uniformly Expanding Systems 59 4.2.3 Example 4.2.1 revisited 61 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 63 4.3 Point Processes of Rare Events 64 4.3.1 Absence of Clustering 64 4.3.2 Presence of Clustering 65 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 67 4.4 Conditions Д q (u n ), D 3 (u n ), D p (u n ) * and Decay of Correlations 68 4.5 Specific Dynamical Systems where the Dichotomy Applies 71 4.5.1 Rychlik Systems 72 4.5.2 Piecewise Expanding Maps in Higher Dimensions 73 4.6 Extreme Value Laws for Physical Observables 74

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Section: Non-exponential Lawsmentioning

confidence: 90%

“…While developing the techniques in [143] to sharpen the error terms, it has been necessary to improve the estimates in [49, Proposition 1] (this is done in Proposition 4.1.12 below). One consequence of this is that the authors were then able to essentially remove condition SP p,θ (u n ).…”

Section: The New Conditionsmentioning

confidence: 99%

Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

Self Cite

194

253

View full text Add to dashboard Cite

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“…where the scaling exponent of 1/2 matches the exponent in the spike of the invariant density. Such relations follow from O'Brien's formula for the extremal index (see [FFT2,(2.6)] for a dynamical setting of this), and given the connection between extremal indices and scaling limits for escape rates established in [BDT], we conjecture that it holds in greater generality for scaling limits.…”

Section: Geometric Potentialsmentioning

confidence: 58%

Asymptotic escape rates and limiting distributions for multimodal maps

Demers¹,

Todd²

2020

Ergod. Th. Dynam. Sys.

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We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and nonperiodic points, as the diameter of the hole goes to zero.

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“…This idea already appeared in the context of extreme value theory in [9] (see Proposition 1 and Theorem 1) and in [11] (see Proposition 2.7), where the extremal index is shown to characterize also the clustering of maxima in a stochastic process.…”

Section: Introductionmentioning

confidence: 90%

Almost sure convergence of the clustering factor in α-mixing processes

Abadi

Saussol

2016

Stoch. Dyn.

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Abadi and Saussol (2011) have proved that the first time a dynamical system, starting from its equilibrium measure, hits a target set A has approximately an exponential law. These results hold for systems satisfying the α-mixing condition with rate function α decreasing to zero at any rate. The parameter of the exponential law is the product λ(A)µ(A), where the latter is the measure of the set A; only bounds for λ(A) were given. In this note we prove that, if the rate function α decreases algebraically and if the target set is a sequence of nested cylinders sets An(x) around a point x, then λ(An) converges to one for almost every point x. As a byproduct, we obtain the corresponding result for return times.

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Speed of convergence for laws of rare events and escape rates

Cited by 35 publications

References 43 publications

Extremes and Recurrence in Dynamical Systems

Extremes and Recurrence in Dynamical Systems

Asymptotic escape rates and limiting distributions for multimodal maps

Almost sure convergence of the clustering factor in α-mixing processes

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