Extremes of Lévy-driven spatial random fields with regularly varying Lévy measure

Rønn-Nielsen, Anders; Stehr, Mads

doi:10.1016/j.spa.2022.04.007

Cited by 4 publications

(5 citation statements)

References 22 publications

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“…for almost all realizations (y v ) of (Y v ), which is also satisfied for the approximating fields (Z t v ) for each t ∈ N. The proof follows by arguments similar to those of [26], however with some modifications due to lack of stationarity caused by conditioning on (Y v ) = (y v ). This is handled using the ergodic-like convergence of Lemma A.1 in Appendix A.…”

Section: Appendix C: Extremal Representationmentioning

confidence: 64%

“…Results on extremes of stationary random fields are to the best of the authors' knowledge mostly formulated under the assumption of (D n ) being a sequence of increasing boxes ( [13,20,24,29]). However, there are recent papers in which extremes are considered under a more general geometric regime; see for instance [23,25,26]. In this paper we simply assume that D n ⊆ Z d can be approximated from the in-and outside by unions of certain slowly increasing boxes, where the approximations asymptotically have the same size as D n ; please see Assumption 1 for details.…”

Section: Introductionmentioning

confidence: 99%

“…It is well known in the literature that a regularly varying random field satisfying certain mixing and anti-clustering conditions has a well defined extremal index that furthermore is determined by its spectral distribution; see the seminal paper [7] for the one-dimensional regularly varying case and [26,29] for conditions ensuring an extremal index for random fields that are not necessarily regularly varying. In [21] a one-dimensional stochastic volatility model of the form X t = Y t Z t is studied, where (Z t ) is regularly varying and (Y t ) is i.i.d.…”

Section: Introductionmentioning

confidence: 99%

“…The second cluster counting process, similar to that of [26], is relatively new in the context of extreme value theory. The process is based on the more intuitive rule that a cluster comprises of the indices in Φ n which are relatively close to one another.…”

Section: Introductionmentioning

confidence: 99%

“…Assumption 1 can be seen as a relaxation of the geometrical assumption on the sequence of index sets in [26,Section 2], where a certain type of convexity is required. We let be an arbitrary translation invariant total order on Z d (e.g.…”

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confidence: 99%

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Extremes of regularly varying stochastic volatility fields

Stehr¹,

Rønn-Nielsen²

2023

Preprint

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We consider a stationary stochastic volatility field YvZv with v ∈ Z d , where Z is regularly varying and Y has lighter tails and is independent of Z. We make-relative to existing literature-very general assumptions on the dependence structure of both fields. In particular this allows Y to be nonergodic, in contrast to the typical assumption that it is i.i.d., and Z to be given by an infinite moving average.Considering the stochastic volatility field on a (rather general) sequence of increasing index sets, we show the existence and form of a Y -dependent extremal functional generalizing the classical extremal index. More precisely, conditioned on the field Y , the extremal functional shows exactly how the extremal clustering of the (conditional) stochastic volatility field is given in terms of the extremal clustering of the regularly varying field Z and the realization of Y .Secondly, we construct two different cluster counting processes on a fixed, full-dimensional set with boundary of Lebesgue measure zero: By means of a coordinate-dependent upscaling of subsets, we systematically count the number of relevant clusters with an extreme observation. We show that both cluster processes converge to a Poisson point process with intensity given in terms of the extremal functional.

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Section: Appendix C: Extremal Representationmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Extremes of regularly varying stochastic volatility fields

Stehr¹,

Rønn-Nielsen²

2023

Preprint

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Extremes of subexponential Lévy-driven random fields in the Gumbel domain of attraction

Stehr

Rønn-Nielsen

2021

Extremes

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Extremes for stationary regularly varying random fields over arbitrary index sets

Passeggeri,

Wintenberger

2024

Extremes

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We consider the clustering of extremes for stationary regularly varying random fields over arbitrary growing index sets. We study sufficient assumptions on the index set such that the limit of the point processes of the exceedances above a high threshold exists. Under the so-called anti-clustering condition, the extremal dependence is only local. Thus the index set can have a general form compared to previous literature (Basrak and Planinić in Bernoulli 27(2):1371–1408, 2021; Stehr and Rønn-Nielsen in Extremes 24(4):753–795, 2021). However, we cannot describe the clustering of extreme values in terms of the usual spectral tail measure (Wu and Samorodnitsky in Stochastic Process Appl 130(7):4470–4492, 2020) except for hyperrectangles or index sets in the lattice case. Using the recent extension of the spectral measure for star-shaped equipped space (Segers et al. in Extremes 20:539–566, 2017), the $$\Upsilon$$ Υ -spectral tail measure provides a natural extension that describes the clustering effect in full generality.

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Extremes of Lévy-driven spatial random fields with regularly varying Lévy measure

Cited by 4 publications

References 22 publications

Extremes of regularly varying stochastic volatility fields

Extremes of regularly varying stochastic volatility fields

Extremes of subexponential Lévy-driven random fields in the Gumbel domain of attraction

Extremes for stationary regularly varying random fields over arbitrary index sets

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