Domain-scaled regular variation: Mathematical foundations for a new tail approximation

Kirstin Strokorb; Marco Oesting; Raphaël de Fondeville

doi:10.1016/j.spa.2026.105004

Domain-scaled regular variation: Mathematical foundations for a new tail approximation  

Kirstin Strokorb
, Marco Oesting
, Raphaël de Fondeville

Department of Mathematical Sciences

Universität Stuttgart
Federal Statistical Office

Research output: Contribution to journal › Article › peer-review

Abstract

Threshold exceedances of stochastic processes in space and time often appear to be more localized the more extreme they are. While classical regularly varying stochastic processes cannot model this effect, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate this behaviour. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown–Resnick process and turns out to be natural in this context. We study key properties of the resulting tail approximating process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness.

Original language	English
Article number	105004
Number of pages	30
Journal	Stochastic Processes and their Applications
Volume	200
Early online date	3 Jun 2026
DOIs	https://doi.org/10.1016/j.spa.2026.105004
Publication status	E-pub ahead of print - 3 Jun 2026

Acknowledgements

The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for the kind hospitality during our two weeks at the institute in March 2024 supported through the program “Oberwolfach Research Fellows”, which has allowed us to kickstart this research. KS wishes to thank Cardiff University for granting her research leave during autumn 2024, which helped to establish the first version of this manuscript, including financial support for visiting MO at the University of Stuttgart during this time. We would like to thank an anonymous referee for several insightful comments to improve the quality of this manuscript. Importantly, this allowed us to correct the conditions in Lemma 2.14 in the second version of [1], and we include a counterexample in Remark 2.15 therein similar to the one suggested by the referee. While both of these arguments are no longer needed for the present version, as we restrict our attention solely on the point evaluation functional satisfying (A.7), we owe it to the referee to rethink the presentation of this material in a more streamlined manner including further links to relevant literature.

Keywords

Asymptotic independence
Brown-Resick process
Domain-scaling
Gaussian process
Max-stability
Pareto process
Poisson point process
Regular variation
Tail process
Threshold exceedance
Triangular array

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Cite this

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title = "Domain-scaled regular variation: Mathematical foundations for a new tail approximation  ",

abstract = "Threshold exceedances of stochastic processes in space and time often appear to be more localized the more extreme they are. While classical regularly varying stochastic processes cannot model this effect, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate this behaviour. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown–Resnick process and turns out to be natural in this context. We study key properties of the resulting tail approximating process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness.",

keywords = "Asymptotic independence, Brown-Resick process, Domain-scaling, Gaussian process, Max-stability, Pareto process, Poisson point process, Regular variation, Tail process, Threshold exceedance, Triangular array",

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T1 - Domain-scaled regular variation: Mathematical foundations for a new tail approximation  

AU - Strokorb, Kirstin

AU - Oesting, Marco

AU - de Fondeville, Raphaël

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Y1 - 2026/6/3

N2 - Threshold exceedances of stochastic processes in space and time often appear to be more localized the more extreme they are. While classical regularly varying stochastic processes cannot model this effect, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate this behaviour. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown–Resnick process and turns out to be natural in this context. We study key properties of the resulting tail approximating process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness.

AB - Threshold exceedances of stochastic processes in space and time often appear to be more localized the more extreme they are. While classical regularly varying stochastic processes cannot model this effect, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate this behaviour. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown–Resnick process and turns out to be natural in this context. We study key properties of the resulting tail approximating process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness.

KW - Asymptotic independence

KW - Brown-Resick process

KW - Domain-scaling

KW - Gaussian process

KW - Max-stability

KW - Pareto process

KW - Poisson point process

KW - Regular variation

KW - Tail process

KW - Threshold exceedance

KW - Triangular array

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DO - 10.1016/j.spa.2026.105004

M3 - Article

SN - 0304-4149

VL - 200

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

M1 - 105004

ER -

Domain-scaled regular variation: Mathematical foundations for a new tail approximation

Abstract

Acknowledgements

Keywords

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Cite this

Domain-scaled regular variation: Mathematical foundations for a new tail approximation  