Entry and return times distribution

Haydn, Nicolai

doi:10.1080/14689367.2013.822459

Cited by 38 publications

(32 citation statements)

References 67 publications

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“…The later point is an interesting application of the general fact that, for many hyperbolic dynamical systems, the hitting time of small balls, once renormalized, converges in distribution to an exponential random variable (see e.g. the reviews [16,56,29]). Once we have tightness and convergence in distribution, we can evaluate the moments of N p .…”

Section: 3mentioning

confidence: 99%

Potential kernel, hitting probabilities and distributional asymptotics

Pène¹,

Thomine²

2019

Ergod. Th. Dynam. Sys.

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Z d -extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green-Kubo's formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve in turn the asumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on abelian covers of compact manifolds of negative curvature are discussed. arXiv:1702.06625v2 [math.DS] 16 May 2017 POTENTIAL KERNEL, HITTING PROBABILITIES AND DISTRIBUTIONAL ASYMPTOTICS 2 observables need only to decay polynomially at infinity, instead of having bounded support. We apply it to the geodesic flow on abelian covers of compact manifolds with negative curvature.This article is organized as follow. We present our setting and our results in Section 1, as well as our applications to Lorentz gases (Sub-subsection 1.4.1) and to geodesic flows (Sub-subsection 1.4.2). In Section 2 we present our spectral assumptions, and prove Theorem 1.4 using the method of moments. In Section 3 we prove Theorems 1.7 and 1.11, and in Section 4 the two applications mentioned above. We discuss Green-Kubo's formula in the Appendix.

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Section: 3mentioning

confidence: 99%

Potential kernel, hitting probabilities and distributional asymptotics

Pène¹,

Thomine²

2019

Ergod. Th. Dynam. Sys.

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“…Parallel to the extreme value theory and totally independently, it was deeply studied the theory of hitting times in Poincaré Recurrence Theory. The review papers [5,11,14] bring a major panorama of classical results. Hitting times to balls and cylinder sets were specifically considered.…”

Section: Introductionmentioning

confidence: 99%

Clustering indices and decay of correlations in non-Markovian models

Abadi

Freitas

2019

Nonlinearity

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When there is no independence, abnormal observations may have a tendency to appear in clusters instead of scattered along the time frame. Identifying clusters and estimating their size are important problems arising in statistics of extremes or in the study of quantitative recurrence for dynamical systems. In the classical literature, the Extremal Index appears associated to the cluster size and, in fact, it is usually interpreted as the reciprocal of the mean cluster size. This quantity involves a passage to the limit and in some special cases this interpretation fails due to an escape of mass when computing the limiting point processes. Smith [18] introduced a regenerative process exhibiting such disagreement. Very recently, in [3] the authors used a dynamical mechanism to emulate the same inadequacy of the usual interpretation of the Extremal Index. Here, we consider a general regenerative process that includes Smith's model and show that it is important to consider finite time quantities instead of asymptotic ones and compare their different behaviours in relation to the cluster size. We consider other indicators such as what we call the sojourn time, which corresponds to the size of groups of abnormal observations, when there is some uncertainty regarding where the cluster containing that group was actually initiated. We also study the decay of correlations of the non-Markovian models considered.

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“…There is currently much interest in understanding the entry and return time distributions of various classes of dynamical systems, in particular in the case of small target sets; cf. [1,11,17,18,20,19,25,26,27,32,43] and references therein. These statistics may be viewed as quantitative refinements of the classical Poincaré recurrence, as pointed out by Kac in his study of return times for ergodic maps and discrete stochastic processes [28].…”

Section: Introductionmentioning

confidence: 99%

Entry and return times for semi-flows

Marklof

2017

Nonlinearity

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Abstract. Haydn, Lacroix and Vaienti [Ann. Probab. 33 (2005)] proved that, for a given ergodic map, the entry time distribution converges in the small target limit, if and only if the corresponding return time distribution converges. The present note explains how entry and return times can be interpreted in terms of stationary point processes and their Palm distribution. This permits a generalization of the results by Haydn et al. to non-ergodic maps and continuous-time dynamical systems.

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Entry and return times distribution

Cited by 38 publications

References 67 publications

Potential kernel, hitting probabilities and distributional asymptotics

Potential kernel, hitting probabilities and distributional asymptotics

Clustering indices and decay of correlations in non-Markovian models

Entry and return times for semi-flows

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