Clustering indices and decay of correlations in non-Markovian models

Abadi, Miguel; Freitas, Ana Cristina Moreira; Freitas, Jorge Milhazes

doi:10.1088/1361-6544/ab37b8

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…Regenerative processes. Here we give two examples, one which exhibits some pathology and which was also recently used in [3] and another one to show that nearly all compound Poisson distributions can be achieved.…”

Section: 1mentioning

confidence: 99%

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Limiting entry times distribution for arbitrary null sets SETS

Haydn,

Vaienti

2019

Preprint

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We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the cluster sizes, where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity. In the special case of periodic points we recover the known Pólya-Aeppli distribution which is associated with geometrically distributed cluster sizes. We apply this method to several examples the most important of which is synchronisation of coupled map lattices. For the invariant absolutely continuous measure we establish that the returns to the diagonal is compound Poisson distributed where the coefficients are given by certain integrals along the diagonal.

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Section: 1mentioning

confidence: 99%

“…To emphasise the regularity condition made in Theorem 2 we look at an example by Smith [24] which was also recently used in [2,3] to exhibit some pathology. Let Y j for j ∈ Z be i.i.d.…”

Section: Smith Examplementioning

confidence: 99%

Limiting entry times distribution for arbitrary null sets SETS

Haydn,

Vaienti

2019

Preprint

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“…We remark that, in the counterexample built by Smith in [28], one can show that the regenerative process is also mixing the behaviour of an EI equal to 0 with that of an EI equal to 1, which we defer to [3]. Moreover, in all counterexamples, the EI still coincides with reciprocal of the limit of the mean finite time cluster size distribution.…”

Section: Introductionmentioning

confidence: 71%

Dynamical counterexamples regarding the Extremal Index and the mean of the limiting cluster size distribution

Abadi,

Freitas,

Freitas

2018

Preprint

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The Extremal Index is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an Extremal Index for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non periodic point. The occurrence of extreme events is then tied to the entrance and recurrence to the vicinities of those points. This enables to mix the behaviour of an Extremal Index equal to 0 with that of an Extremal Index larger than 0. Using bi-dimensional point processes we explain how mass escapes in order to destroy the usual relation. We also perform a study about the formulae to compute the limiting cluster size distribution introduced in [14, 4] and prove that ergodicity is enough to establish that the reciprocal of the Extremal Index is equal to the limit of the mean of the finite time cluster size distribution, which, in the case of the counterexamples given, does not coincide with the mean of the limit of the cluster size distribution.

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“…We remark that, in the counterexample built by Smith in [30], one can show that the regenerative process is also combining the behaviour of an EI equal to 0 and an EI equal to 1, which we defer to [3]. Moreover, in all counterexamples, the EI still coincides with the reciprocal of the limit of the means of the finite time cluster size distributions.…”

Section: Introductionmentioning

confidence: 80%

Dynamical counterexamples regarding the extremal index and the mean of the limiting cluster size distribution

Abadi

Freitas

2020

J. London Math. Soc.

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The extremal index (EI) is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an EI for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non-periodic point. The occurrence of extreme events is then tied to the entrance and recurrence to the vicinities of those points. This enables to mix the behaviour of an EI equal to 0 with that of an EI larger than 0. Using bi-dimensional point processes, we explain how mass escapes in order to destroy the usual relation. We also perform a study about the formulae to compute the cluster size distribution introduced earlier and prove that ergodicity is enough to establish that the finite versions of the reciprocal of the EI and of the mean of the cluster size distribution do coincide. Contents

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Clustering indices and decay of correlations in non-Markovian models

Cited by 6 publications

References 17 publications

Limiting entry times distribution for arbitrary null sets SETS

Limiting entry times distribution for arbitrary null sets SETS

Dynamical counterexamples regarding the Extremal Index and the mean of the limiting cluster size distribution

Dynamical counterexamples regarding the extremal index and the mean of the limiting cluster size distribution

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