Fourier coefficients of invariant random fields on the sphere (and more...)

Recently some attention has been attracted to the investigation of random fields on the sphere. This mostly because of the need of producing a reasonable model for the analysis and the statistical estimation of the data of the Cosmic Microwave Background, a cosmic radiation whose behavior gives important information on the Big-Bang. The most natural approach is to develop a random field into a Fourier series with respect to a natural orthonormal basis of the square integrable functions on the sphere (the spherical harmonics). One of the typical issues of the Big-Bang theory is that this cosmic radiation should be rotationally invariant (i.e. that its distribution is invariant by rotation). In this talk we first consider the problem of characterizing the joint distribution of the Fourier coefficients of a random field which is rotationally invariant. In particular we shall prove that if the Fourier coefficients are independent the they are necessarily Gaussian (and the random field is Gaussian too). Consequences, extensions and applications will follow.