Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure

Research output: Working paperResearch

Abstract

We consider a space-time random field on Rd × R given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time-axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time-point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time-interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

Original languageEnglish
Place of PublicationAarhus
PublisherCentre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University
Number of pages34
Publication statusPublished - 2019
SeriesCSGB Research Reports
Number9

Keywords

  • Convolution equivalence
  • Infinite divisibility
  • Lévy-based modelling
  • Asymptotic equivalence
  • Sample paths for random fields

Cite this

Stehr, M., & Rønn-Nielsen, A. (2019). Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure. Aarhus: Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University. CSGB Research Reports, No. 9
Stehr, Mads ; Rønn-Nielsen, Anders . / Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure. Aarhus : Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University, 2019. (CSGB Research Reports; No. 9).
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abstract = "We consider a space-time random field on Rd × R given as an integral of a kernel function with respect to a L{\'e}vy basis with a convolution equivalent L{\'e}vy measure. The field obeys causality in time and is thereby not continuous along the time-axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying L{\'e}vy measure. Particular examples are the asymptotic probability that there is a time-point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time-interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.",
keywords = "Convolution equivalence, Infinite divisibility, L{\'e}vy-based modelling, Asymptotic equivalence, Sample paths for random fields, Convolution equivalence, Infinite divisibility, L{\'e}vy-based modelling, Asymptotic equivalence, Sample paths for random fields",
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Stehr, M & Rønn-Nielsen, A 2019 'Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure' Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University, Aarhus.

Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure. / Stehr, Mads; Rønn-Nielsen, Anders .

Aarhus : Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University, 2019.

Research output: Working paperResearch

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T1 - Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure

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N2 - We consider a space-time random field on Rd × R given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time-axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time-point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time-interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

AB - We consider a space-time random field on Rd × R given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time-axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time-point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time-interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

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KW - Asymptotic equivalence

KW - Sample paths for random fields

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Stehr M, Rønn-Nielsen A. Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure. Aarhus: Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University. 2019.