35 pages with 2 figuresThe main point of this paper is the investigation of invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in space, with quasi-exponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the high frequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets
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