scholarly journals Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure

2016 ◽  
Vol 53 (1) ◽  
pp. 244-261 ◽  
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Random Field ◽  
Large Class ◽  
Kernel Function ◽  
Random Fields ◽  
Tail Asymptotics ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Asymptotic Probability ◽  
The Right

Abstract We consider a continuous, infinitely divisible random field in Rd given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields we compute the asymptotic probability that the supremum of the field exceeds the level x as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

scholarly journals Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

2017 ◽  
Vol 54 (3) ◽  
pp. 833-851 ◽  
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Random Field ◽  
Large Class ◽  
Random Fields ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Excursion Sets ◽  
Excursion Set ◽  
Fixed Radius ◽  
Asymptotic Probability ◽  
The Right

Abstract We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure

2021 ◽  
Vol 58 (1) ◽  
pp. 42-67 ◽  
Author(s):  
Mads Stehr ◽  
Anders Rønn-Nielsen
Keyword(s):  
Space Time ◽  
Time Interval ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Time Model ◽  
Spatial Object ◽  
Tail Behaviour ◽  
Fixed Radius ◽  
The Right ◽  
Space Time Model

AbstractWe consider a space-time random field on ${{\mathbb{R}^d} \times {\mathbb{R}}}$ 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.

scholarly journals Central limit theorem for mean and variogram estimators in Lévy–based models

2019 ◽  
Vol 56 (01) ◽  
pp. 209-222
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Kernel Function ◽  
Limit Theorems ◽  
Regularity Conditions ◽  
Infinitely Divisible ◽  
The Mean ◽  
The Moment

AbstractWe consider an infinitely divisible random field in ℝd given as an integral of a kernel function with respect to a Lévy basis. Under mild regularity conditions, we derive central limit theorems for the moment estimators of the mean and the variogram of the field.

scholarly journals Identification and isotropy characterization of deformed random fields through excursion sets

2018 ◽  
Vol 50 (3) ◽  
pp. 706-725
Author(s):  
Julie Fournier
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Random Fields ◽  
Weak Form ◽  
Invariance Property ◽  
Basic Domain ◽  
Excursion Sets ◽  
The Mean ◽  
Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

scholarly journals Estimation and filtering of fractional generalised random fields

2000 ◽  
Vol 69 (3) ◽  
pp. 336-361 ◽  
Author(s):  
J. M. Angulo ◽  
M. D. Ruiz-Medina ◽  
V. V. Anh
Keyword(s):  
Brownian Motion ◽  
Random Field ◽  
Least Squares ◽  
Random Fields ◽  
General Class ◽  
Duality Theory ◽  
Weak Sense ◽  
Linear Estimation ◽  
Special Cases ◽  
Estimation And Filtering

AbstractThis paper considers the estimation and filtering of fractional random fields, of which fractional Brownian motion and fractional Riesz-Bessel motion are important special cases. A least-squares solution to the problem is derived by using the duality theory and covariance factorisation of fractional generalised random fields. The minimum fractional duality order of the information random field leads to the most general class of solutions corresponding to the largest function space where the output random field can be approximated. The second-order properties that define the class of random fields for which the least-squares linear estimation problem is solved in a weak-sense are also investigated in terms of the covariance spectrum of the information random field.

A Bayesian Approach for Detection of Isotropy Violation of a Random Field Over a Sphere

2017 ◽  
Vol 69 (1) ◽  
pp. 64-70
Author(s):  
Tarun Souradeep ◽  
Santanu Das ◽  
Benjamin D. Wandelt
Keyword(s):  
Random Field ◽  
Quantitative Evaluation ◽  
Bayesian Approach ◽  
Random Fields ◽  
Spherical Harmonic ◽  
Covariance Structure ◽  
General Covariance ◽  
General Method

We present a general method for estimating the isotropy violation of random fields on a sphere using a Bayesian formalism. We use Bipolar Spherical Harmonic (BipoSH) representation of general covariance structure on the sphere. Our approach promises to provide a robust quantitative evaluation of the evidence for SI violation-related anomalies in the CMB sky by estimating the BipoSH spectra along with their complete posterior.

scholarly journals Anisotropic scaling of the random grain model with application to network traffic

2016 ◽  
Vol 53 (3) ◽  
pp. 857-879 ◽  
Author(s):  
Vytautė Pilipauskaitė ◽  
Donatas Surgailis
Keyword(s):  
Network Traffic ◽  
Random Fields ◽  
Covariance Function ◽  
Asymptotic Form ◽  
Scaling Limits ◽  
Infinitely Divisible ◽  
Area Distribution ◽  
Anisotropic Scaling ◽  
Grain Model ◽  
Heavy Tailed

AbstractWe obtain a complete description of anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian, and ‘intermediate’ infinitely divisible random fields. The asymptotic form of the covariance function of the random grain model is obtained. Application to superimposed network traffic is included.

scholarly journals Fractional random fields associated with stochastic fractional heat equations

2005 ◽  
Vol 37 (01) ◽  
pp. 108-133 ◽  
Author(s):  
M. Ya. Kelbert ◽  
N. N. Leonenko ◽  
M. D. Ruiz-Medina
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Evolution Equations ◽  
Heat Equations ◽  
Mean Square ◽  
Fractional Evolution Equations ◽  
The Mean

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

Sign in / Sign up

Export Citation Format

Share Document