Cosmic Microwave Background (CMB) Anisotropies is a subject of intensive research in recent years, and therefore it is necessary to develop suitable theory and methods for the analysis of isotropic fields on spheres. The main object of our paper is to show that the polyspectra can be given as the coefficients of the orthogonal expansion of cumulants of the field in terms of irreducible tensor products of spherical harmonics. We obtain necessary and sufficient conditions for isotropy of a

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... ssian field and the conditions are stated in terms of higher order spectra (polyspectra). The relation between cumulants and spectra gives a new method of estimating spectra. MSC 2010 subject classifications: Primary 60G60, 60K35; secondary 60K35