How to compute the extremal index of stationary random fields

Ferreira, Helena; Pereira, Luísa

doi:10.1016/j.spl.2007.11.025

Cited by 24 publications

(13 citation statements)

References 4 publications

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“…Let the assumptions of the theorem be satisfied. Then (8) holds. Dividing the set {M p(n) > v n } into p * (n) = p 1 (n)p 2 (n) · · · p d (n) disjoint sets and applying monotonicity and stationarity we obtain that…”

Section: Main Theoremmentioning

confidence: 96%

On maxima of stationary fields

Soja-Kukieła

2019

J. Appl. Probab.

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Let {Xn : n ∈ Z d } be a weakly dependent stationary random field with maxima MA := sup{X i : i ∈ A} for finite A ⊂ Z d and Mn := sup{X i : 1 ≤ i ≤ n} for n ∈ N d . In a general setting we prove that P(M (N 1 (n),N 2 (n),...,N d (n)) ≤ vn) = exp(−n d P(X0 > vn, MA n ≤ vn)) + o(1) for some increasing sequence of sets An of size o(n d ), where (N1(n), N2(n), . . . , N d (n)) → (∞, ∞, . . . , ∞) and N1(n)N2(n) · · · N d (n) ∼ n d . The sets An are determined by a translation invariant total order on Z d . For a class of fields satisfying a local mixing condition, including m-dependent ones, the main theorem holds with a constant finite A replacing An. The above results lead to new formulas for the extremal index for random fields. The new method of calculating limiting probabilities for maxima is compared with some known results and applied to the moving maximum field.

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Section: Main Theoremmentioning

confidence: 96%

On maxima of stationary fields

Soja-Kukieła

2019

J. Appl. Probab.

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“…So it is rather surprising that there seems to be no existing limit theory for m-dependent stationary random fields. To the authors' knowledge the only attempts in this direction have been made by Turkman (2006) and Ferreira and Pereira (2008), but their results do not yield any concise method of calculating the limits and the extremal index for m-dependent random fields. In particular, the formula proposed in the latter paper does not work for the simple 1-dependent random field given in Example 5.5 below.…”

Section: Introductionmentioning

confidence: 98%

Managing local dependencies in asymptotic theory for maxima of stationary random fields

Jakubowski

Soja-Kukieła

2018

Extremes

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In the paper we solve the limit problem for partial maxima of m-dependent stationary random fields and we extend the obtained solution to fields satisfying some local mixing conditions. New methods for describing the limitting distribution of maxima are proposed. A notion of a phantom distribution function for a random field is investigated. As an application, several original formulas for calculation of the extremal index are provided. Moving maxima and moving averages as well as Gaussian fields satisfying the Berman condition are considered.

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“…then θ is called the extremal index. To the computation of the extremal index of a random field in N 2 , the paper [13] is devoted; in [14], the asymptotic location of the maximum of a random field with a certain extremal index was studied.…”

Section: Introductionmentioning

confidence: 99%

Extremal Indices in a Series Scheme and Their Applications

2015

Inf.Appl.

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We generalize the concept of extremal index of a stationary random sequence to the series scheme of identically distributed random variables with random series sizes tending to infinity in probability. We introduce new extremal indices through two definitions generalizing the basic properties of the classical extremal index. We prove some useful properties of the new extremal indices. We show how the behavior of aggregate activity maxima on random graphs (in information network models) and the behavior of maxima of random particle scores in branching processes (in biological population models) can be described in terms of the new extremal indices. We also obtain new results on models with copulas and threshold models. We show that the new indices can take different values for the same system, as well as values greater than one.

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How to compute the extremal index of stationary random fields

Cited by 24 publications

References 4 publications

On maxima of stationary fields

On maxima of stationary fields

Managing local dependencies in asymptotic theory for maxima of stationary random fields

Extremal Indices in a Series Scheme and Their Applications

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