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

Stehr, Mads; Rønn-Nielsen, Anders

doi:10.1007/s10687-021-00428-0

Cited by 2 publications

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“…The next example illustrates how Assumption 2.2 looks like when C n is connected and expands at the same pace in all directions. This assumption was the one used in the papers [10,13,14].…”

Section: Geometric Assumption and Preliminariesmentioning

confidence: 99%

“…In the literature, results for spatial objects, comparable to some of the present results, are to the best of the authors' knowledge only formulated under the assumption of (D n ) n∈N being a sequence of increasing boxes; see for instance [5,12] that provides results like the ones found in Section 3 below under this additional geometrical assumption. In contrast, we allow the index sets D n to expand in a much more general way; we refer to the authors' papers [10,13,14] for similar however slightly less general assumptions on the sequence of index sets, but with other and less general results. Our requirement will simply be that each D n is the lattice points of a sufficiently nice full-dimensional subset of R d .…”

Section: Introductionmentioning

confidence: 99%

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Extremal clustering and cluster counting for spatial random fields

Rønn-Nielsen¹,

Stehr²

2022

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We consider a stationary random field indexed by an increasing sequence of subsets of Z d obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail distribution of the individual variables relates to the tail behavior of the maximum of the field over the index sets in the limit as the index sets expand.Furthermore, in a framework where we let the increasing index sets be scalar multiplications of a fixed set C, potentially with different scalars in different directions, we use two cluster definitions to define associated cluster counting point processes on the rescaled index set C; one cluster definition divides the index set into more and more boxes and counts a box as a cluster if it contains an extremal observation. The other cluster definition that is more intuitive considers extremal points to be in the same cluster, if they are close in distance. We show that both cluster point processes converge to a Poisson point process on C. Additionally, we find a limit of the mean cluster size. Finally, we pay special attention to the case without clusters.

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“…The next example illustrates how Assumption 2.2 looks like when C n is connected and expands at the same pace in all directions. This assumption was the one used in the papers [10,13,14].…”

Section: Geometric Assumption and Preliminariesmentioning

confidence: 99%