1,721,064 research outputs found
Fluctuations of Level Curves for Time-Dependent Spherical Random Fields
Full text linkThe investigation of the behaviour for geometric functionals of random fields
on manifolds has drawn recently considerable attention. In this paper, we extend this framework
by considering fluctuations over time for the level curves of general isotropic Gaussian spherical
random fields. We focus on both long and short memory assumptions; in the former case, we
show that the fluctuations of u-level curves are dominated by a single component, corresponding
to a second-order chaos evaluated on a subset of the multipole components for the random field
The stochastic properties of l(1)-regularized spherical Gaussian fields
Full text linkConvex regularization techniques are now widespread tools for solving inverse problems in a variety of different frameworks. In some cases, the functions to be reconstructed are naturally viewed as realizations from random processes; an important question is thus whether such regularization techniques preserve the properties of the underlying probability measures. We focus here on a case which has produced a very lively debate in the cosmological literature, namely Gaussian and isotropic spherical random fields, and we prove that neither Gaussianity nor isotropy are conserved in general under convex regularization based on ℓ1ℓ1 minimization over a Fourier dictionary, such as the orthonormal system of spherical harmonics
The Needlet CMB Trispectrum
No full textWe propose a computationally feasible estimator for the needlet trispectrum, which develops earlier work on the bispectrum by Donzelli et al. (2012). Our proposal seems to enjoy a number of useful properties, in particular a) the construction exploits the localization properties of the needlet system, and hence it automatically handles masked regions; b) the procedure incorporates a quadratic correction term to correct for the presence of instrumental noise and sky-cuts; c) it is possible to provide analytic results on its statistical properties, which can serve as a guidance for simulations. The needlet trispectrum we present here provides the natural building blocks for the efficient estimation of nonlinearity parameters on CMB data, and in particular for the third order constants g(N) (L) and tau(N) (L)
Approximate normality of high-energy hyperspherical eigenfunctions
No full textThe Berry heuristic has been a long standing ansatz about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see [7]). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace–Beltrami operator on the normalized d-dimensional sphere – also known as spherical harmonics). Some applications to non-Gaussian models are also discussed
U-Statistics on the spherical poisson space
No full textWe review a recent stream of research on normal approximations for linear functionals and more general U-statistics of wavelets/needlets coefficients evaluated on a homogeneous spherical Poisson field. We show how, by exploiting results from Peccati and Zheng (Electron J Probab 15(48):1487-1527, 2010) based on Malliavin calculus and Stein's method, it is possible to assess the rate of convergence to Gaussianity for a triangular array of statistics with growing dimensions. These results can be exploited in a number of statistical applications, such as spherical density estimations, searching for point sources, estimation of variance, and the spherical two-sample problem
Spherical Poisson Waves
Full text linkWe introduce a model of Poisson random waves in and we study
Quantitative Central Limit Theorems when both the rate of the Poisson process
and the energy (i.e., frequency) of the waves (eigenfunctions) diverge to
infinity. We consider finite-dimensional distributions, harmonic coefficients
and convergence in law in functional spaces, and we investigate carefully the
interplay between the rates of divergence of eigenvalues and Poisson governing
measures
A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions
Full text linkA lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data
On the limiting behaviour of needlets polyspectra
Full text linkThis paper provides quantitative Central Limit Theorems for nonlinear transforms of spherical random fields, in the high-frequency limit. The sequences of fields that we consider are represented as smoothed averages of spherical Gaussian eigenfunctions and can be viewed as random coefficients from continuous wavelets/needlets; as such, they are of immediate interest for spherical data analysis. In particular, we focus on so-called needlets polyspectra, which are popular tools for non-Gaussianity analysis in the astrophysical community, and on the area of excursion sets. Our results are based on Stein-Malliavin approximations for nonlinear transforms of Gaussian fields, and on an explicit derivation on the high-frequency limit of their variances, which may have some independent interest
Flexible-bandwidth Needlets
Full text linkWe investigate here a generalized construction of spherical wavelets/needlets which admits extra-flexibility in the harmonic space, i.e., it allows the corresponding support in multipole (frequency) space to vary in more general forms than in the standard constructions. We study the analytic properties of this system and we investigate its behaviour when applied to isotropic random fields: more precisely, we establish asymptotic localization and uncorrelation properties (in the high-frequency sense) under broader assumptions than typically considered in the literature
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