### Abstract

We consider a space-time random field on R* ^{d}* × 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 language | English |
---|---|

Place of Publication | Aarhus |

Publisher | Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University |

Number of pages | 34 |

Publication status | Published - 2019 |

Series | CSGB Research Reports |
---|---|

Number | 9 |

### Keywords

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

### Cite this

*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

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

Research output: Working paper › Research

TY - UNPB

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

AU - Stehr, Mads

AU - Rønn-Nielsen, Anders

PY - 2019

Y1 - 2019

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.

KW - Convolution equivalence

KW - Infinite divisibility

KW - Lévy-based modelling

KW - Asymptotic equivalence

KW - Sample paths for random fields

KW - Convolution equivalence

KW - Infinite divisibility

KW - Lévy-based modelling

KW - Asymptotic equivalence

KW - Sample paths for random fields

M3 - Working paper

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

PB - Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Aarhus University

CY - Aarhus

ER -