Abstract
We consider the statistical analysis of random sections of a spin fibre bundleover the sphere. These may be thought of as random fields that at each point $p\in \mathbb{S}^{2}$ take as a value a curve (e.g. an ellipse) living in the tangent plane at that point $T_{p}\mathbb{S}^{2}$, rather than a number as in ordinary situations. The analysis of such fields is strongly motivated by applications, for instance polarization experiments in Cosmology. To investigate such fields, spin needlets were recently introduced by [21] and [20]. We consider the use of spin needlets for spin angular power spectrum estimation, in the presence of noise and missing observations, and we provide Central Limit Theorem results, in the high frequency sense; we discuss also tests for bias and asymmetries with an asymptotic justification.
Highlights
The analysis of functions defined on the sphere by means of wavelets has recently been the object of a number of theoretical and applied papers, see for instance [3], [4], [5, 61], [41, 42], [22, 23, 24], [7, 8], [27]
Concerning astrophysics, there are many mathematical papers which have been motivated by the analysis of so-called Cosmic Microwave Background radiation (CMB); the latter can be very loosely viewed as a relic electromagnetic radiation which permeates the Universe providing a map of its status from 13.7 billions ago, in the immediate adjacency of the Big Bang
Almost all mathematical statistics papers in this area have been concerned with the temperature component of CMB, which can be represented as a standard spherical random field
Summary
The analysis of (random or deterministic) functions defined on the sphere by means of wavelets has recently been the object of a number of theoretical and applied papers, see for instance [3], [4], [5, 61], [41, 42], [22, 23, 24], [7, 8], [27] Many of these works have found their motivating rationale in recent developments in the applied sciences, such as Medical Imaging, Geophysics, Atmospheric Sciences, Astrophysics and Cosmology. In more formal terms (to be explained later), this can be summarized by saying that we shall be concerned with random sections of fibre bundles over the sphere; from a more group-theoretic point of view, we shall show that polarization random fields are related to so-called spin-weighted representations of the group of rotations SO(3). Throughout this paper, given two positive sequences {aj} , {bj} we shall write aj ≈ bj if there exist positive constants c1, c2 such that c1aj ≤ bj ≤ c2aj for all j ≥ 1
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