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We study the representations of tensor random fields on the sphere basing on the theory of representations of the rotation group. Introducing specific components of a tensor field and imposing the conditions of weak isotropy and mean square continuity, we derive their spectral decompositions in terms of generalized spherical functions. The properties of random coefficients of the decompositions are characterized, including such an important question as conditions of Gaussianity.

Abstract. Many statistical applications require establishing central limit theorems for sums/integrals Mathematics Subject Classification. 60F05, 62M10, 60G15, 62M15, 60G10, 60G60.