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High-frequency asymptotics for Lipschitz–Killing curvatures of excursion sets on the sphere

Vadlamani, Sreekar LU and Marinucci, Domenico (2016) In Annals of Applied Probability 26(1). p.462-506
Abstract
In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler–Poincaré characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the investigation of asymmetries... (More)
In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler–Poincaré characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the investigation of asymmetries and anisotropies in cosmological data. The statistics we focus on are also suitable to deal with spherical random fields which can only be partially observed, the canonical example being provided by the masking effect of the Milky Way on Cosmic Microwave Background (CMB) radiation data.

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author
publishing date
type
Contribution to journal
publication status
published
subject
keywords
high-frequency asymptotics, spherical random fields, Minkowski functionals, excursion sets, Gaussian subordination, 60G60, 62M15, 53C65, 42C15
in
Annals of Applied Probability
volume
26
issue
1
pages
45 pages
publisher
Institute of Mathematical Statistics
external identifiers
  • scopus:84958693876
ISSN
1050-5164
DOI
10.1214/15-AAP1097
language
English
LU publication?
no
id
f9948ed9-92ca-416b-ad35-fca5e3915ca5
date added to LUP
2017-09-01 11:55:16
date last changed
2017-09-03 05:26:41
@article{f9948ed9-92ca-416b-ad35-fca5e3915ca5,
  abstract     = {In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler–Poincaré characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the investigation of asymmetries and anisotropies in cosmological data. The statistics we focus on are also suitable to deal with spherical random fields which can only be partially observed, the canonical example being provided by the masking effect of the Milky Way on Cosmic Microwave Background (CMB) radiation data.<br/><br/>},
  author       = {Vadlamani, Sreekar and Marinucci, Domenico},
  issn         = {1050-5164},
  keyword      = {high-frequency asymptotics,spherical random fields,Minkowski functionals,excursion sets,Gaussian subordination,60G60,62M15,53C65,42C15},
  language     = {eng},
  number       = {1},
  pages        = {462--506},
  publisher    = {Institute of Mathematical Statistics},
  series       = {Annals of Applied Probability},
  title        = {High-frequency asymptotics for Lipschitz–Killing curvatures of excursion sets on the sphere},
  url          = {http://dx.doi.org/10.1214/15-AAP1097},
  volume       = {26},
  year         = {2016},
}