The Annals of Applied Probability

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

Domenico Marinucci and Sreekar Vadlamani

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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.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 462-506.

Dates
Received: February 2014
Revised: November 2014
First available in Project Euclid: 5 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1452003245

Digital Object Identifier
doi:10.1214/15-AAP1097

Mathematical Reviews number (MathSciNet)
MR3449324

Zentralblatt MATH identifier
1334.60089

Subjects
Primary: 60G60: Random fields 62M15: Spectral analysis 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 42C15: General harmonic expansions, frames

Keywords
High-frequency asymptotics spherical random fields Gaussian subordination Lipschitz–Killing curvatures Minkowski functionals excursion sets

Citation

Marinucci, Domenico; Vadlamani, Sreekar. High-frequency asymptotics for Lipschitz–Killing curvatures of excursion sets on the sphere. Ann. Appl. Probab. 26 (2016), no. 1, 462--506. doi:10.1214/15-AAP1097. https://projecteuclid.org/euclid.aoap/1452003245


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