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

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.

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

“…As an application of the previous result, let us consider the Fourier components normalized to have variance one; the GKF yields immediately (compare Marinucci and Vadlamani (), Corollary 5, see also Cheng and Xiao ()).and…”

A note on global suprema of band-limited spherical random functions

“…see [16]. For fixed j, as in (2) it follows from results in [1] that there exist α > 1 and µ + > 0 such that, for all u > µ + P sup…”

“…Spherical random fields have recently drawn a lot of applied interest, especially in an astrophysical environment (see [6], [14]); closed form expressions for the density of their maxima and for excursion probabilities have been given in ( [10], [9], [16]). In particular, the latter references exploit the Gaussian Kinematic Fundamental formula by Adler and Taylor (see [1]) to approximate excursion probabilities by means of the expected value of the Euler-Poincarè characteristic for excursion sets.…”

“…denotes the derivative of the covariance function at the origin, see again ( [10], [9], [16]). When working on compact domains as the sphere, it is often of great interest to focus on sequences of band-limited random fields; for instance, a very powerful tool for data analysis is provided by fields which can be viewed as a sequence of wavelet transforms (at increasing frequencies) of a given isotropic spherical field T. More precisely, take b(.)…”

“…Intuitively, sample paths become rougher and rougher as j grows, hence any fixed threshold is crossed with probability tending to one. In [16], uniform bounds for band-limited fields have indeed been established, covering even nonGaussian circumstances; however these bounds require a further averaging in the space domain for the fields considered, and this averaging ensures the uniform boundedness of λ j ; in these circumstances, the multiplicative constant on the right-hand side of (3) can be simply incorporated into the exponential choosing a different constant 1 < α ′ < α. There is, however, a question that naturally arises for the cases where λ j diverges -e.g., whether it is possible to provide bounds on global suprema, allowing the thresholds to grow with frequency.…”