Excursion sets and non-Gaussian void statistics

Guido D’Amico; Marcello Musso; Jorge Noreña; Aseem Paranjape

doi:10.1103/physrevd.83.023521

Correlation of excursion sets for non-Gaussian cosmic microwave background temperature distributions

Monthly Notices of the Royal Astronomical Society ◽

10.1046/j.1365-8711.1998.01399.x ◽

1998 ◽

Vol 296 (3) ◽

pp. 693-700 ◽

Cited By ~ 16

Author(s):

R. B. Barreiro ◽

J. L. Sanz ◽

E. Martínez-González ◽

J. Silk

Keyword(s):

Cosmic Microwave Background ◽

Microwave Background ◽

Excursion Sets ◽

Temperature Distributions ◽

Cosmic Microwave Background Temperature ◽

Background Temperature ◽

Non Gaussian

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On the excursion area of perturbed Gaussian fields

ESAIM Probability and Statistics ◽

10.1051/ps/2020002 ◽

2020 ◽

Vol 24 ◽

pp. 252-274

Author(s):

Elena Di Bernardino ◽

Anne Estrade ◽

Maurizia Rossi

Keyword(s):

Limit Theorem ◽

Random Perturbation ◽

Wasserstein Distance ◽

Expansion Formula ◽

Kinematic Formula ◽

Small Perturbations ◽

Asymptotically Normal ◽

Excursion Sets ◽

Growing Domain ◽

Non Gaussian

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2 under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

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Excursion sets of three classes of stable random fields

Advances in Applied Probability ◽

10.1239/aap/1275055229 ◽

2010 ◽

Vol 42 (2) ◽

pp. 293-318 ◽

Cited By ~ 11

Author(s):

Robert J. Adler ◽

Gennady Samorodnitsky ◽

Jonathan E. Taylor

Keyword(s):

Random Fields ◽

Gaussian Fields ◽

Mean Values ◽

Geometric Characteristics ◽

Excursion Sets ◽

Asymptotic Formulae ◽

The Mean ◽

Non Gaussian ◽

Stable Random Fields

Studying the geometry generated by Gaussian and Gaussian-related random fields via their excursion sets is now a well-developed and well-understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves.

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Excursion sets of three classes of stable random fields

Advances in Applied Probability ◽

10.1017/s0001867800004067 ◽

2010 ◽

Vol 42 (02) ◽

pp. 293-318 ◽

Cited By ~ 6

Author(s):

Robert J. Adler ◽

Gennady Samorodnitsky ◽

Jonathan E. Taylor

Keyword(s):

Random Fields ◽

Gaussian Fields ◽

Mean Values ◽

Geometric Characteristics ◽

Excursion Sets ◽

Asymptotic Formulae ◽

The Mean ◽

Non Gaussian ◽

Stable Random Fields

Studying the geometry generated by Gaussian and Gaussian-related random fields via their excursion sets is now a well-developed and well-understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves.

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